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A  R  I  T  II  M  r.  1  i  ^ 


PRODUCTIVE     SYSTEM 

ACCOMPANIED   BY    A 

KEY 


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•J    IKODUCTIVE    0 


I  SREOTYPE    EDlTIOii 


HARTFORD 

PUBLISHJ^D  BY  JOHN  PAINE- 


I 


LIBRARY 


University  of  California, 


GIF^T   OK 


Mrs.  SARAH  P.  WALSWORTH 

Received  October,  i8g4. 
^Accessions  No.  ^  ^1^1  Cp  •      Class  No. 


Smitl)'0  JTctD  2lrUl)mttic. 
ARITHMETIC 


PRODUCTIVE     SYSTEM 


ACCOMPANIKD   BY  A 


KEY 


CUBICAL    BLOCKS 


BY  ROSWELL  C.  SMITH, 


AVTHOR   OF   PRACTICAL  AND   MENTAL  ARITHMETIC,   THE   PRODVCTIVB   QBAMMAB, 
THE  PRODUCTIVE   GSOaBAPHY,  &C. 


STEEEOTYPE    EDITION 


HARTFORD : 

PUBLISHED  BY  JOHN  PAINE, 

1842. 


"  IT  IS  NOT  EASY  TO  DEVISE  A  CURE  FOR  SUCH  A  STATE  OF  THINGS,  (tHE 
DECLINING  TASTE  FOR  SCIENCE  ;)  BUT  THE  MOST  OBVIOUS  REMEDY  IS  TO  PRO- 
VIDE THE  EDUCATED  CLASSES  WITH  A  SERIES  OF  WORKS  ON  POPULAR  AN1> 
PRACTICAL  SCIENCES,  FREED  FROM  MATHEMATICAL  SYMBOLS  AND  TECHNI 
CAL  TERMS,  WRITTEN  IN  SIMPLE  AND  PERSPICUOUS  LANGUAGE,  AND  ILLUS- 
TRATED BY  FACTS  AND  EXPERIMENTS  WHICH  ARE  LEVEL  TO  THE  CAPACITY 
OF  ORDINARY  MINDS."  QUARTERLY  ReVIEW. 


nr/t 


"the  first  thing  to  be  required  in  a  system  of  popular  instruc- 
tion, is,  that  it  should  be  intelligible  ;  that  children  and  youth 
should  understand  what  they  learn.  understand  what  they 
learn  ?  it  may  be  asked  ;  what  else  can  they  do  1  we  answer,  that 
they  may  commit  it  to  memory,  may  recite  it,  may  even  make  a  fair 
show  op  knowledge,  and  yet  know  nc thing.  we  have  not  the 
least  hesitation  in  saying,  that  two  or  three  years,  in  the  edu 
cation  of  almost  every  individual  in  this  country,  have  been 
thrown  away  upon  studying  what  they  did  not  understand." 

North  American  Review. 


Entered  according  to  Act  of  Congress,  in  the  year  1841, 

BY   JOHN   PAINE, 

in  the  Clerk's  Office  of  the  District  Court  of  Connecticut, 


HARTFORD, 

STEREOTYPED   BT 

RICHARD    H.    HOBBS. 


ARITHMETIC/ 


PART    FIRST: 


BBINO  A  MIBNTAL2  COURSE  FOR  EVERY  CLASS  OF  LEARNERS. 

NUMERATION.' 

II3P  Recite  by  the  Questions. 

I.     1.   Number,  which  shows  how  many  are  meant,  is  represent- 
ed* by  letters,  by  words,  and  by  characters*  called  figures,  as : — 

One  hundred c 100 

Two  hundred cc 200 

Three  hundred ccc 300 

Four  hundred cccc 400 

Five  hundred d  or  lot 500 

Six  hundred dc  or  loc 600 

Seven  hundred  ....  doc  or  locc 700 

Eight  hundred dccc  or  loccc ....  800 

Nine  hundred dcccc  or  locccct .  .  900 

One  thousand m  or  ciot 1000 

Two  thousand li  or  mm 2000 

Three  thousand   ....  iii  or  mmm 3000 

Four  thousand rv  or  mmmm  ....  4000 

Five  thousand v  or  loo 5000 

Six  thousand vl  or  ioom 6000 

Seven  thousand  ....  vli  or  ioomm  ....  7000 

Eight  thousand vm  or  ioommm  . .  .  8000 

Nine  thousand fx  or  ioommmm  .  .  .  9000 

Ten  thousand x  or  ccioot  ....  10000 

Twenty  thousand  .  .  .1x  or  cciooccioo.  20000 

Fifty  thousand l  or  looo 50000 

Onehundredthousand. c or ccciooo  .  .  100000 
Five  hundred  thousand.  i5  orioooo.  .  .  .  500000 
One  million m or ccccioooo.  1000000 


One  .  .  . 

.  I  .  . 

. .  1 

Two.  .  . 

.  II   . 

.2 

Three.  . 

.  Ill  . 

.  .3 

Four.  .  . 

.  IV*. 

.  .4 

Five .  .  . 

.  V.  . 

..5 

Six.  .  .  . 

.  VI  . 

..6 

Seven .  . 

.  VII. 

..7 

Eight  .  . 

.  VIII 

..8 

Nine.  .  . 

.  IX*. 

.  .9 

Ten  .  .  . 

.  X    .  . 

.10 

Eleven  . 

.  XI    . 

.  11 

Twelve  . 

.  XII. 

.  12 

Thirteen 

.  xin 

.  13 

Fourteen 

.  XIV 

.  14 

Fifteen  . 

.  XV  . 

.  15 

Sixteen . 

.  XVI. 

.  16 

Twenty. 

.  XX  . 

.20 

Thirty.  . 

.  XXX 

.30 

Forty  .  . 

.  XL* 

.40 

Fifty.  .  . 

.  L  .  . 

.50 

Sixty  .  . 

LX   . 

.60 

Seventy  . 

LXX. 

.  70 

Eighty  . 

LXXX 

.80 

Ninety  . 

xc* 

.  90 

*  Or,  mi  for  4;  vim  for  9  ;  xxxx  for  40 ;  lxxxx  for  90  ;  and  cm  for  900, 
t  Every  o  annexed  to  lo  increases  its  value  10  times  :  as  lo,  is  500,  loo  is  5000 ;  in  like 
manner  the  prefixing  of  c  and  ttie  annexing  of  o  to  cio  increases  it  10  times  as  cio,  1000, 
ccioo,  10000;  lastly  aline  over  any  number  increases  it  1000  times:  as,  d,  500,  d,  600000. 

Note.  L.  stands  for  the  Latin  language  ;  G.  for  the  Greek  and  F.  for  the  French. 
1  Arithmetic,  [G.  arithmetike.'i  Computing,  calculating  or  reckoning  by  numbers. 
S  Mental,  fL.  mentis.']  Pertaining  to  the  mind;  intellectually. 

3  Numeration,  [L.  numeratio.'i  Numbering ;  the  method  or  act  of  numbering. 

4  Represented,    Exhibited ;  described ;  personated ;  to  supply  the  place  of. 

5  Character,    A  mark;  a  stamp;  a  letter;  reputation;  a  personage. 


4  MENTAL    ARITHMETIC. 

QUESTIONS   ON   THE    FOREGOING. 

2.  What  does  Number  show?  1.  How  is  it  represented?  1. 
What  letters  and  what  figures  stand  for  one,  four,  five  and  nine! 
What  for  ten?     Ans.  The  figure  1  and  0  called  naught  or  cipher. 

3.  What  letters  and  what  figures  stand  for  eleven]  For  twelve? 
Fifteen?  Twenty?  Forty?  Fifty?  Ninety?  What  different  numbers 
may  be  represented  by  the  figures  1  and  5  written  together  I  Ans.  Fif- 
teen and  fifty-one ;  as,  15:51. 

4.  What  different  numbers  may  be  expressed  by  the  figures  1  and 
9  ?  2  and  5  ?  What  number  is  expressed  by  c  ?  by  d  ?  by  m  ?  What 
by  c,  D,  and  m,  with  a  dash  over  each. 

5.  What  figures  stand  for  one  hundred ?  For  two  hundred?  One 
thousand?  Ten  thousand?  One  hundred  thousand ?  One  million? 


ADDITION/ 

QUESTIONS. 

II.     1.  Thomas  has  3  dollars  and  Rufus  5  dollars.     How  many 
dollars  have  they  both?  Say  3  and  5  are  8.  A.  8  dollars. 

2.  A  farmer  has  5  cows  in  his  yard,  and  6  in  the  pasture.     How 
many  cows  has  he  in  both  places? 

3.  A  man  bought  a  hat  for  5  dollars,  and  a  pair  of  boots  for  7  dol- 
lars.    How  many  dollars  did  he  pay  for  both  ? 

4.  A  man  lost  7  dollars,  and  then  had  8  dollars  left?     How  many 
dollars  had  he  at  first  ? 

5.  A  man  gave  8  dollars  for  a  saddle,  5  dollars  for  a  bridle,  and  2 
dollars  for  a  whip.     What  did  the  whole  cost  him? 

6.  Suppose  there  are  8  oranges  in  a  basket,  5  on  a  table,  and  4  in 
my  pockets.     How  many  will  they  all  make  ? 

7.  A  grocer  sold  to  one  man  6  barrels  of  flour,  to  another  9  barrels, 
and  still  had  5  barrels  left.     How  many  barrels  had  he  at  first? 

8.  In  a  certain  class  are  6  large  boys,  7  small  ones,  and  10  girls. 
How  many  scholars  are  there  in  the  class? 

9.  A  boy  has  10  dollars,  his  father  gave  him  10  more,  and  he  has 
5  owing  to  him.     How  many  will  they  all  make  ? 

10.  Thomas  read  19  pages  of  history  in  one  day,  12  in  another, 
and  9  in  another.     How  many  pages  did  he  read  in  all? 

11.  Suppose  you  are  10  years  old,  and  that  your  brother  was  10 
years  old  when  you  were  born.     What  is  his  age  now  ? 

12.  A  man  gave  10  dollars  more  for  his  horse  than  for  his  wagon, 
and  the  wagon  cost  him  30  dollars.     What  did  they  both  cost  him? 

13.  How  many  are  10  and  30  and  40?  60  and  10  and  20?  100 
and  200  and  400?  600  and  300  and  100? 

1  Addition,    [L.  additio.l    Any  thing  added;  adding;  joining;   uniting  two  or 
more  numbers  in  one  sum. 


ADDITION. 


iiiiiim 

iiiiiiiiiiiiiii- 

iiiiiHiiiiiiimiriK 

IIIIIIIIIIIIIIIIIIIIIIIIMIIII- 


mil 


IIIIIIIIMIIIII- 


iiii 


-  3 

'  e 

10 
K 

-21 
28 
96 

■45 
JS 


iiniiiiMuiiitiii 


iiiiiiiiiiiiiiiiiii 

iiiiiiiuiiiiiiiiiniiiiitiii 


14.  When  it  is  1  o'clock,  a  regular  clock  strikes  once ;  when  it  is 
2  o'clock,  twice ;  when  it  is  3  o'clock,  thrice.  How  many  strokes 
will  these  make?     A.  6  strokes :  because  1  and  2  are  3,  and  3  are  6. 

15.  Add  together  all  the  strokes  that  a  clock  strikes  in  12  hours, 
as  in  the  above  picture  of  a  clock.  [Thus  1  and  2  are  3,  and  3  are  6, 
and  4  are  10,  and  so  on  up  to  12.] 

In  the  following  table  the  single  figures  are  combined^  by  pairs  in 
every  possible  manner ;  so  that,  if  it  be  committed  to  memory,  the 
learner  can  add  with  facility^  any  two  numbers  whatever. 


ADDITION      TABLE. 

and 
and 

and 
and 

and 
and 

and 
and 

and 

1 
2 

3 

4 
5 

6 
7 
8 
9 

are 

are 

or 
or 

or 
or 

or 
or 

or 

2 
3 

2 
3 

and   9    0 
and    9    0 

r   3 

r   4 

4 
5 

and    8    or   4 
and    8    or    5 

and 
and 

and 
and 

and 
and 

7    ( 

7  c 

8  c 

8  c 

9  c 
9  c 

r    5 

r    6 

r    6 

r    7 

r    7 

r    8 

8 

9 

and 
and 

and 
and 

and 

and 

and 
and 

6 
6 

7 
7 
8 
8 
9 
9 

are    11 
arc    12 

are    13 
are    I4 

are    15 
are    16 

are    17 

are    18 

2a 

2a 

,d2 
.d3 

are    4 
are    5 

or    3  an 
or    3  an 

or    3  an 
or     3  an 

or    3  an 

and    9    or    5 
and    9    or    6 

2  and  4 
2  and  5 

2  and  6 

2  and  7 

d  3 

d  4 
A  5 

d  6 

J  7 

are    Q 

are    7 

6 

7 

or    4  ai 
or    4  a 

or    4  an 

.d4 
Id  5 

I  6 

are 
are 

or 

8 
9 

5   a 

2ar 

>d8 

.d5 

are 



.  .    10 

16.  Repeat  the  above  table  beginning  at  the  top  on  the  left.  What 
are  all  the  different  pairs  of  single  figures  that  together  make  7 1 
make  9?  10?  12?  13?  14^  15?  16?  17?  18? 

17.  How  many  are  8  and  4  ?  18  and  4  ?  28  and  4?  48  and  4?  7 
and  6  ?  17  and  6  ?  26  and  7  ?  57  and  6  ?  86  and  7  ?  96  and  7  ? 

18.  How  many  are  9  and  5  ?  19  and  5  ?  55  and  9  ?  8  and  8  ?  38 
and  8?  98and8?  5and7?  65and7?  9and8?  49and8?  99and8? 

19.  Add  together  audibly  MO  twos:  10  threes:  10  fours:  10  fives: 
10  sixes  :   10  sevens  :  10  eights  :  10  nines  :  10  tens. 

20.  How  many  are  78  and  10  and  3  ?  78  and  13  ?  91  and  14  [10 
and  4]  ?  105  and  15  [10  and  5]  ?  120  and  16  ?  136  and  17  ?  153  and 
18?  171  and  19? 

21.  Add  together  audibly'  all  the  different  numbers  under  20,  be- 
ginning with  the  lowest. 

1  Combined,    United  closely;  associated;  leagued;  confederated. 

2  Facilitt,  [L./oci/ifoj.]    Easiness  to  be  performed ;  readiness  ;  affability. 

3  Audibly,    In  a  manner  so  as  to  be  beard ;  in  an  audibto  manner. 

1* 


MENTAL   ARITHMETIC. 


SUBTRACTION/ 

QUESTIONS. 

III.  1.  A  boy  having  15  cents  lost  10;  how  many  had  he  left? 
Say  10  from  15  leaves  5  ;  because  10  and  5  are  15.       A.  5  cents. 

2.  A  man  owing  12  dollars  paid  7  dollars.  How  many  dollars  did 
he  still  owe  1  7  from  12  leaves  how  many  and  why  ? 

3.  A  grocer  bought  a  barrel  of  molasses  for  15  dollars,  and  sold  it 
for  18  dollars.     How  much  did  he  make  on  it"? 

4.  Suppose  your  age  to  be  12  years,  and  your  brother's  20  years. 
What  is  the  difference  between  your  age  and  his  1 

5.  A  farmer  bought  a  coW  for  20  dollars  and  sold  her  for  15  dol- 
lars.    What  was  his  loss  1 

6.  Thomas  and  William  counted  their  nuts ;  the  former  had  30 
and  the  latter  50.     How  many  had  one  more  than  the  other  1 

7.  A  certain  cistern  has  one  pipe  by  which  25  gallons  run  in  every 
hour,  and  another  pipe  by  which  19  gallons  run  out  every  hour.  How 
many  gallons  will  stay  in  every  hour  1 

8.  How  many  are  left  in  taking  3  from  1113  from  21 1  3  from  51  % 
4  from  13  ?  4  from  23  ?  4  from  53  T  4  from  83  1 

9.  How  many  does  5  from  12  leave  1  5  from  321  6  from  16?  6 
from  36  ?  7  from  15  ?  9  from  17  ?  9  from  37 1  9  from  107 1 

10.  How  many  cents  added  to  100  cents  will  make  200  cents? 

11.  What  is  the  difference  between  300  cents  and  700  cents? 

12.  How  much  smaller  is  70  than  270  ?  500  than  1000  ? 

13.  How  much  larger  is  590  than  90  ]   1275  than  275  ? 

14.  A  gentleman  paid  250  dollars  for  his  carriage,  and  50  dollars 
less  for  his  horse.     What  was  the  price  of  his  horse  ? 


MULTIPLICATION." 

QUESTIONS. 

IV.  1.  A  farmer  gave  10  dollars  for  a  calf,  10  dollars  for  a  plough 
and  10  dollars  for  a  load  of  hay.     What  did  he  pay  for  the  whole  ? 

Say  10  and  10  and  10  are  30 ',  Or  as  10  is  taken  3  times,  rather  say 
at  once,  3  times  10  are  30,  as  in  the  Table.  A,  30  dollars. 

2.  What  will  5  hats  cost  at  4  dollars  a-piece  ?        A.   20  dollars. 

3.  A  man  bought  10  sheep  for  3  dollars  a-piece.  What  did  he  pay 
for  the  whole  ?  How  many  are  3  times  10  ? 

4.  If  a  shoemaker  can  manufacture  4  pair  of  shoes  in  one  day, 
how  many  can  he  make  at  that  rate  in  8  days  ?  In  10  days  ? 

1  Subtraction,  [L.  subiractio.}    The  act  of  taking  a  part  fVom  the  rest. 
3  MuiiTiPLiCATioir.    imuUiplicatio.'i  The  act  of  increasing  iq  number. 


MtJLTlPLlCATlOI^* 


5.  If  5  yards  of  cloth  will  make  a  suit  of  clothes,  how  many  yards 
will  it  require  to  make  2  suits'?  8  suits  1  11  suits'?  Repeat  the  fol- 
lowing multiplication  table  : — ' 

MULTIPLICATION      TABLE. 


2 

3 

4 

5 

0 

7 

§     ,       9 

10 

times 

times 

times 

times 

times 

times 

times 

times 

times 

lare2 

1  .ire  3 

lare  4 

lare  5 

lare  6 

lare  7 

lare  8 

1  are    9 

1  are  10 

2  .    4 

2  .    G 

2  .    8 

2  .  10 

2  ,12 

2  .14 

2  .16 

2  .    18 

2  .    20 

3  .    6 

3  .    9 

3  .12 

3  .  15 

3  .  18 

3  .21 

3  .24 

3  .    27 

3  .    30 

4  .    8 

4  .  12 

4  ,  16 

4  .20 

4  .24 

4  .28 

4  .32 

4  .    36 

4  .    40 

5  .10 

5  .  15 

5  .20 

5  .25 

5  .30 

5  .35 

5  -40 

5  ,    45 

5  .    50 

6  .  12 

6  .  18 

6  .24 

6  .30 

6  .36 

6  .42 

6  .48 

6  .    54 

6  .    60 

7  .14 

7  .21 

7  ,28 

7  .35 

7  .42 

7  .49 

7  .50 

7  .    63 

7.    70 

8  .16 

8  ,24 

8  .32 

8  .40 

8  .48 

8  ,56 

8  .64 

8  .    72 

8  .    80 

9  .  18 

9  .27 

9  .36 

9  .45 

9  .54 

9  .63 

9  .72 

9  .    81 

9  ,    90 

10  .20 

10  .30 

10  .40 

10  .50 

10  .60 

10  .70 

10  .80 

10  .    90 

10  .100 

11  .22 

11  .  33 

11  .44 

11  .55 

11  .  66 

11  .77 

11  .88 

11  .    99 

11  .110 

12  .24 

12  .36 

12  .48 

12  .60 

12  ,72 

12  .84 

12  .96 

12  .  108 

12  ,  120 

11 

18                         1 

times 

times      times 

times 

times 

times 

times        times 

1  are  11 

4are44    7are77 

10  are  110 

1  are  12 

4  are  48 

7areP4     10  are  120 

2  .    22 

5  .    55   8  .    88 

11  ,    121 

2  ,     24 

5  .    60 

8  .    96    11  .    132 

3  .    33  1  6  .    66  j  9  .    99 

12  .    132 

3  .    36    6  .    72 1 

9  .  108    12  .    144 

6.  It  takes  4  pecks  to  make  1  bushel.     How  many  pecks  then  are 
there  in  3  bushels  1  In  5  bushels '?  In  7  bushels  1 

7.  At  6  cents  a  quart,  what  will  6  quarts  of  cherries  cost  1  What 
will  8  quarts  cosf?  9 quarts'?   10  quarts?  11  quarts'?  12  quarts'? 

8.  When  the  board  of  a  small  family  costs  7  dollars  a  week,  what 
will  the  board  for  7  weeks  cost '?  For  8  weeks  1 

9.  When  flour  is  8  dollars  a  barrel,  how  much  must  you  pay  for  8 
barrels'?  For  9  barrels]  For  11  barrels'?  For  12  barrels! 

10.  A  man  traveled  by  stage,  at  the  rate  of  7  miles  an  hour.     How 
far  did  he  go  in  7  hours "?  In  9  hours '?  In  12  hours  1 

11.  When  hay  is  9  dollars  a  ton,  what  will  be  the  cost  of  7  tons'? 
9  tons t  8  tons?  6  tons? 

12.  If  10  men  can  do  a  job  of  work  in  10  days,  how  long  will  it 
take  one  man  alone  to  do  it,  working  at  the  same  rate  ? 

13.  How  many  are  6  times  3  ?  9  times  5  ?  8  times  7  ?  7  times  6  ? 
8  times  6  ?  12  times  5  ?  9  times  12  ? 

14.  If  a  cannon  ball  fly  at  the  rate  of  11  miles  a  minute,  how  far 
would  it  go  in  7  minutes  ?  In  9  minutes  ?  In  12  minutes  ? 

15.  How  many  are  10  times  5?  10  times  10?  10  times  20?  10 
times  30  ?   10  times  40  ?  10  times  90  ?  10  times  100  ? 

16.  If  a  regiment  consists  of  1000  men,  of  how  many  men  would 
2  regiments  consist  ?  3  regiments  ?  5  regiments  ? 


MENTAL    ARITHMETIC. 


DIVISION/ 


QUESTIONS. 

v.     1.  How  many  yards  of  cloth  at  2  dollars  a  yard  can  you  buy 
for  8  dollars,  and  why  1         A.  4  yards  ;  because  4  times  2  are  8. 

2.  How  many  yards  of  cloth  at  3  dollars  a  yard  can  you  buy  for 
12  dollars?  For  18  dollars'?  For  24  dollars?  For  30  dollars? 

3.  At  5  dollars  a  hat,  how  many  hats  will  40  dollars  buy?     As 
many  hats  as  5  is  contained  times  in  40.  A.  8  hats. 

4.  A  father  having  24  books,  gave  3  to  each  of  his  children.    How 
many  children  must  he  have  had  ? 

5.  A  merchant  bought  a  quantity  of  flour  for  144  dollars,  paying 
12  dollars  a  barrel.     How  many  barrels  did  he  buy  ? 

6.  If  one  man  alone  can  perform  a  piece  of  work  in  100  days,  how 
long  would  it  take  10  such  men  to  do  the  same  ? 

7.  If  a  man  can  travel  6  miles  in  an  hour  by  stage,  how  long  will  it 
take  him  to  perform  a  journey  of  72  miles?  Of  GO  miles? 

8.  Suppose  an  orchard  to  have  132  trees  in  rows,  with  12  trees  in 
a  row,  of  how  many  rows  does  the  orchard  consist  ? 

9.  How  many  times  8  in  96  ?  9  in  63  ?  8  in  56  ?  9  in  108  ?    11  in 
132?   12  in  108?   12  in  132? 

10.  A  man  having  300  dollars,  gave  to  his  sons  100  dollars  a-piece. 
How  many  sons  had  he  ?  How  many  times  100  in  300  ? 

11.  How  many  times  100  in  500  ?   5  in  500  ?  200  in  1000  ?   4  in 
800?  6  in  1200?  10  in  1000?   1000  in  10000? 

12.  If  1  barrel  holds  5  bushels  of  rye ;  how  many  barrels  will  21 
bushels  fill  ?  A.  4  barrels  and  1  bushel  over. 

13.  How  many  times  5  in  42,  and  how  many  over  ?  6  in  39,  and 
how  many  over  ?  7  in  46  ?  8  in  74  ?  9  in  98  ?  9  in  99  ? 

14.  Why  not  say  9  in  99  only  10  times  and  9  over  ?    A.  Because 
9  is  contained  in  99  all  of  1 1  times. 

15.  How  many  times  6  in  41  ?  8  in  102  ?  9  in  112  ?   12  in  155  ^ 


FRACTIONS." 

QUESTIONS. 

VI.  1.  To  divide  13  dollars  equally  among  2  persons ;  how  would 
you  proceed  to  find  each  man's  exact  part?  Say  2  in  13,  6  times 
and  1  dollar  over ;  the  1  over  is  considered  as  divided  into  2  equal 
parts  by  writing  the  2  under  the  1,  with  a  line  between,  making  ^ 
which  is  read  1-half  A.  6^  dollars. 

2.  How  is  any  number  divided  into  2,  3,  4,  5,  &c.  equal  parts  ? 

A.  By  writing  under  it  as  above  the  dividing  numbers,  2,  3, 5,  &c. 

1  Division,  [L.  divisio.'\  Theact  of  dividing  any  thing  into  parts;  the  state  of  being 
divided ;  that  which  divides  or  separates;  partition ;  a  part  of  an  army  or  militia ;  dia- 
union ;  variance. 

2  Fraction.    [L./rociio.]  The  act  of  breaking ;  the  broken  part  of  a  number. 


FRACTIONS.  9 

3.  How  many  times  2  in  15  ?  A.  71  read  7  and  l-half. 

4.  How  many  times  3  in  13  ?  A.  4^,  read  4  and  1-third. 
6.  How  many  times  4  in  35  ?  A.  8|,  read  8  and  3-fourths. 

6.  How  many  times  5  in  39  ?  A.  7^,  read  7  and  4-fifths. 

7.  How  many  times  6  in  41  ]  A.  61,  read  6  and  5-sixths. 

8.  Howmany  times7in  15?  8  in  30?  9  in  95?  10  in  109^  11  in 
1391  12  in  150?   12  in  155? 

9.  From  what  do  Aa/i;e5,^Air^5,&c.,  take  their  names?  A.  From 
the  figure  below  the  line  whether  it  be  large  or  small. 

10.  What  then,  for  example,  is  meant  by  2, 3,  4,  5, 10, 20, 30,  «fec., 
with  the  figure  1  over  each?  A.  l-half  [l] ;  1-third  [l] ;  1-fourth 
[1];   l-fifth[|];   l-tenth[^];   1 -twentieth  [^V]  ;  1 -thirtieth  [^]. 

11.  A  man  havmg  20  sheep  sold  i  of  them.  How  many  did  he 
se^-     „  A.  10  sheep. 

12.  How  much  is  I  of  20  ?     |of22?  A.  10;  11. 

13.  How  much  is  l  of  60  ?     f  of  60  ?  A.  20  ;  40.* 

14.  How  much  is  |-  of  40  ?     f  of  40  ?  A.  10 ;  30." 

15.  A  man  deposited^  in  his  cellar,  in  thifall,  60  bushels  of  ap- 
ples, and  all  except  |  of  them  rotted  during  the  winter.  How  many 
bushels  of  sound  apples  had  he  in  the  spring  ? 

16.  How  much  is  :!  of  60?     fof60?     4  of  60? 

17.  How  much  is  ^  of  72  ?     |of72?     |-of72? 

18.  How  much  is  \  of  14  ?     ^  of  14  ?    ^  of  14  ? 

19.  A  father  gave  his  son  5  oranges,  which  was  |  of  all  he  had. 
How  many  oranges  had  he  ?  [2  times  5.]  A.  10  oranges. 

20.  If  I  of  a  number  is  5,  what  is  that  number  ? 

21.  If  ^  of  a  number  is  3,  what  is  that  number  ? 

22.  If  ^  of  a  number  is  6,  what  is  that  number  ? 

23.  If  -}  of  a  number  is  7,  what  is  that  number  ? 

24.  If  I  of  the  length  of  a  stick  of  timber  is  5  feet,  how  long  is  the 
whole  stick  ? 

25.  A  merchant  having  40  bushels  of  rye,  sold  1  of  it.  How 
many  bushels  had  he  left  ? 

26.  How  many  cents  would  you  have  left,  if  you  had  24,  and 
should  lose  ^  of  them?  How  many  left  if  you  lost  f  of  them? 

27.  Suppose  you  have  20  barrels  of  flour,  and  sell  in  one  day  }  of 
it ;  in  another  day  |  of  it.  How  many  barrels  will  you  sell  in  both 
days,  and  how  many  barrels  will  you  have  left  ? 

28.  When  hay  sells  for  15  dollars  a  load,  and  you  buy  |  of  a  load, 
how  many  more  dollars  will  buy  the  whole  load  ? 

29.  WhatisAof20?  f of20?  }of24?  |of24?  |of24?  |of24? 

30.  If  John  owns  1  of  a  vessel,  and  you  the  rest ;  how  many  thirds 
do  you  own  ?  How  many  thirds  make  the  whole  ? 

31.  Thomas  owns  3^  of  a  factory,  Rufus  f,  and  Charles  the  rest. 
What  part  does  Charles  own  ? 

32.  If  I  of  a  mellon  costs  2  cents,  what  will  a  whole  one  cost  ? 

1  Dbfositetd,  [L.  depositum.l    Laid  down ;  pledged ;  put. 


10  MENTAL   ARITHMETIC. 

33.  When  -|  of  a  bushel  of  wheat  costs  2  shillings  ;  what  would  | 
cost  1  I  or  1  bushel  cost  1  |  cost  1  2  bushels  or  '/  cost  ] 

34.  When  yu  of  a  pew  costs  10  dollars,  how  many  tenths  at  that 
rate  can  be  bought  for  50  dollars'?  for  80  dollars'?    for  100  dollars T 

35.  What  does  ^,  f,  f,  ^,  &c.  of  any  thing  appear  to  mean'?  A.  ^ 
means  1  of  its  2  equal  parts  ;  f  means  2  of  its  3  equal  parts ;  f ,  3  of 
its  4  equal  parts ;  |,  4  of  its  5  equal  parts,  &c. 

36.  Which  then  is  the  greater  fraction  ^  or  ^^  ■?  ^  oi  ^1  ^  or  ^ "? 

37.  How  many  halves,  or  thirds,  or  fourths,  or  fifths,  &c.,  make  1 
whole"?  A.  I  [2-halves,]  or  f  [3-thirds,]  or  |  [4-fourths,]  or  |  [5-fifths.] 

38.  How  many  halves  then  in  2  wholes'?  thirds  in  4  wholes? 
fourths  in  6  wholes  1  fifths  in  8  wholes  1  sixths  in  9  wholes  ? 

39.  How  many  whole  ones  in  |  [4-halves] "?  in  V  [12-thirds]? 
in  V  ?  in  ^-i  1  in  V  ^ 

40.  When  the  upper  figure  is  the  greater  one,  what  does  it  indi- 
cate?    A.  That  it  is  to  be  divided  by  the  lower  one. 

41.  How  many  wholes  in  V -?  Y  ?  V?  Vi  ^    t¥  "? 

42.  How  many  whole  ones  in  |?  [2  in  5].  A.  2|. 

43.  How  many  halves  in  2|  wholes  1  [2  times  2  and  ^].     A.  f . 

44.  How  many  wholes  in  y  ?  [3  in  17].  A.  5f. 

45.  How  many  thirds  in  5f  wholes  1  [3  times  5  and  f].     A.   y . 

46.  How  many  wholes  in  V  ?  ff  ?  tt  '?   T  ?    t¥  ?   W  ^ 

47.  How  many  fifths  in  4  ?  in  4^  ■?  sevenths  in  8^  ■?  ninths  in  10|  ? 
tenths  in  8^'?  twelfths  in  12||'?  twentieths  in  2^1 

48.  James  has  |  of  a  dollar ;  Rufus  f ,  and  Thomas  |.  How  many 
dollars  and  how  many  eighths  over  have  they  all  ■?     J..  2|  dollars. 

49.  Add  together  ^,  f ,  f ,  |,  |,  f ,  ^,  f ,  and  f .  A.  5. 

50.  Add  together  j\,  y^^,  j\,  ■f'^,  j%,  ^,  ^,  and  ^.      A.  4^. 

51.  A  man  having  ^  of  a  ship,  sold  |  of  it.    What  part  had  he  left  ? 

52.  Take  |  from  f ;  /^  from  f| ;  s^  from  §|. 

53.  If  1  gallon  of  molasses  cost  5^  of  a  dollar,  what  will  2  gallons 
cost  *?  3  gallons  cost  1     How  many  gallons  will  cost  1  dollar  1 

54.  How  much  is  2  times  |1  [for  1].  3  times  I"?  [for  1|]. 
4  times  f1  [2|].  5  times  |?  etimes^f?  7  times  |'? 

55.  If  2  bushels  of  oats  cost  f  [6-eighths]  of  a  doUar,  how  many 
eighths  will  buy  1  bushel '?  A.  |.  [3-eighths]. 

56.  How  much  is  I  of  I?   ^off?   ^of-^?   iof|f1    foff^l  | 

^^181      1^^287        1     „fl20') 
01  30   •     f  01  T5  f    To   01  T4  J  • 

57.  At  6  dollars  a  yard,  what  will  2  yards  of  cloth  cost  \  What 
will  I  a  yard  cost '?  What  will  2^  yards  cost  then '?  How  much  then 
is  2^  times  6  ■? 

58.  How  much  is  3|  times  8,  or  3  times  8  and  |^  of  8  ? 

59.  How  much  is  3f  times  4,  or  3  times  4  and  f  of  4  ? 

60.  How  much  is  7f  times  9,  or  7  times  9  and  f  of  9  ? 

61.  How  much  is  8^  times  10 !  8f  times  10 "?  7|  times  la-? 

62.  When  flour  is  12  dollars  a  barrel,  what  will  3^  barrels  cost? 
4f  barrels  cost  ]  8 j  barrels  cost  ?  lOf  barrels  cost  ] 


FEDERAL    MONEY.  || 

VII.    TABLES 
OF   MONEY,    WEIGHTS   AND    MEASURES. 

FEDERAL*     MONEY,' 

1.  Is  the  currency^  or  coin^  of  the  United  States  ' 

}L"^et*f!^^^"^:^-t----jrV--f 

i^^'"^^^ make....i  doUar\     $ 

^^  ^^"^^«    make  ....  1  eagleB  .  .  E. 

ENGLISH,  OR  STERLING^  MONEY 

2.  Is  used  by  Great  Britain  and  her  dependencies. '» 

3  farthings..        ;••    S±   I  "^  ^  P^""^^  '  '  f  d- 

4  farthings...            'Se    J  f  ^  P^^^^  '  •  ^  d- 
12  pence^^  ,       ^  ^^"""^ d. 

20  sSlincs- '"^^'    ^  '™^^ s. 

^u  fanmings     ^ake    I  pound" £. 

QUESTIONS. 

^.  What  is  Federal  Money  P  l     Reopaf  tf,P  ToM«      tt 

''3'co''  ?;''"''■  "'  ^*^'^«'a,ing"'meZm '"'"''""^^  '"  '"^'"^  ^"^  ««"ing>  and  thence 

usedasmWyTur^a'^iSfSnl^"'''  '"""'  ''"^P^^'  °^  ^^h^r  metal,  stamped  to  be 
4  Mill  is  so  called  from  millp  T     fnr  irt^n    v. 
6  CENT  is  SO  called  from  7SS;i-L,fi?m  -^^ZlTm  ""'"  """'«  '  "'"''■ 

.nd_Ha^.c.„,,  mads  of  copper  .ha  MiKrtfc;;i."r^-lf  .t'S,'''±l' 

.   n  DE?ES»'^'?.''^?CSf'lTbi!i\f ''«*^^  ""» 1-.  coined  i.. 
.ng  subjee.  ,0.    Dependencies  are  «,°,|  or"S  Sec',',"r™' '  "■=  '"'«  "f  ^ 
La  n&S,?- ^  «-"  /-««^ST^S  r ^aiSTi^Xn. 
for"  rc^c~°'  "  ''--•  "^^  «■=  S-^sh  ,,»,„^,  and  ..  from  ,„e  LaUn  ^r^ 

"'I'yfiS^rT.S'Tit^'^-"'  '■'>"''"■"'.  »«»-«  ".e  sMIiinj piece  had  origi- 
J%lSl."  "  '^°'"  "»  ■'«'"  P-^.  «'<BK  and  me  £  «^  ^  j^^  „,  ^,„ 


12  CLOTH    MEASURE. 


TROY    WEIGHT, 


For  weighing  gold,  silver,  liquors,  bread,  &c. 

24  grains  [gr,] make  1  pennyweight  .  .  .  dwt. 

20  pennyweights  .  .  .  make  1  ounce oz. 

12  ounces make  1  pound lb. 


AVOIRDUPOIS    WEIGHT,^ 

6.  For  weighing  hay,  grain,  groceries,  and  all  coarse  articles. 

16  drams  [dr.]  .  .  .  make  1  ounce oz. 

16  ounces make  1  pound lb. 

25  pounds^ make  1  quarter qr. 

4  quarters make  1  hundred  weight  .  cwt. 

20  hundred  weight  make  1  ton T. 


apothecaries'  weight, 
7.  Is  used  for  mixing,  but  not  for  selling  medicines. 

20  grains  [gr.] make 1  scruple  .  .  9 

3  scruples make 1  dram    •  .  .  3 

8  drams make 1  ounce  •  •  •  f 

12  ounces make .1  pound .  .  .  ib 


QUESTIONS. 

8.  For  what  is  Troy  Weight  used  ?  5.  Repeat  the  Table. 

9.  How  many  pounds  in  63  ounces?     A.  5  lb.  3  oz.  or  5  ^  lb. 

10.  Why  are   3  ounces  the  same  as  ^  of  a  pound?     A.  The 
pound  consists  of  12  equal  parts  called  ounces,  therefore  1  ounce  is 


^   of  a  pound;  2  ounces,  ^■^•,  3  ounces,  j\,  &c. 


1  2 


11.  How  many  ounces  are  there  in  d^^  pounds  ?  in  10  y^  pounds  1 
pennyweights  in  22V  ounces  1 

12.  What  is  the  use  of  Avoirdupois  weight  1  6.  Repeat  the  Table. 

13.  How  many  drams  in  1  ounce  5  drams  I  quarters  in  30  pounds? 
f  1 2^]  pounds  in  2  quarters  10  pounds  ?  quarters  in  2f  hundred  weight  1 
pounds  in  1  hundred  weight,  or  4  quarters  ?  in  20  hundred  weight  ? 

14.  For  what  is  Apothecaries' Weight  used  ?  7.  Repeat  the  Table. 

15.  How  many  drams  in  3|  ounces  ?  grains  in  2  scruples  ?  pounds 
in  155  ounces? 

16.  How  many  pounds  in  27  ounces  ?  [2T2]Ounces  in  100  drams? 
Drams  in  300  scruples  ?   Scruples  in  29  grains  ? 

1  175  Troy  ounces  are  equal  to  192  ounces  Avoirdupois ;  1  lb.  Troy,  to  5760  grains, 
and  I  lb.  Avoirdupois  to  7000  grains.  The  pound  and  ounce  in  Apothecaries'  weight  are 
the  same  as  in  Troy  weight ;  the  only  difference  is  in  their  divisions  and  sub-divisions. 

2  Formerly  28  pounds  were  reckoned  for  a  quarter,  making  112  pounds  to  the  him- 
dred,  but  the  practice  has  become  nearly  obsolete. 


WINE    MEASURlE. 


13 


CLOTH    MEASURE, 

17.  Is  used  for  measuring  goods  sold  by  the  yard,  ell,  &c. 


2^  inches  [in.] make  1  naiL na 

qr. 

«   ^ — — xuaKe  1  yard yd 

I  ^^^^J^^« make  1  Flemish  elh  .  Fl.'e. 


:  ^^'^^, make  1  quarter.'  .  .  .  qr. 

^  quarters make  1  yard.  - 


I  'l^^^f  ^« make  1  English  ell.' '.  E  e 

^  ^^^^ters make  1  French  ell.  .  F^.  e. 


DRY    MEASURE, 

18.  For  dry  goods,  as  grain,  fruit,  roots^  coal,  &c 

LCtiP'^ makelquart qt. 

«  ^"^"^J^ make  1  gaUon.  ...  gal 

J.  r""?:^;  ; make  1  bushel.^   .  .  bu. 

^l  J";^t    make  1  chaldron.  .  ch. 

?  ^""^^f^ make  1  quarter.     .  qr. 

^^"^^*^^« makelload load. 

WINE    MEASURE, 

19.  For  spirits,  mead,  vinegar,  cider,  oil,  honey,  &c. 


t  S'^^  fgi-] make  1  pmt „t 

2  Pmts make  1  quart *  '  t 

. ,  f  'l"^^^^ make  1  gallon .'  .*  ^ 

Slo  gallons make  1  barrel.  .  .  .  m 

^?^^;!«"« makeltierce.  ....'.'.';."  tier 

^f  f  ^^"^ make  1  hogshead *£ 

^^P"7«  • make  1  puncheon ^^i 

1  hogsheads make  1  pipe  or  butt. .  .         p  oi  b 

2  P^P^s  •  •  • make  1  tun ....:..  T. 

QUESTIONS, 

20.  For  what  is  Cloth  Measure  used?  17.    Repeat  the  Table 
infEn^S^KfrLWr'   ^"'^^''"   '"^I^'-isheU, 

22.  For  what  is  Dry  measure  used  1   18.    Repeat  the  Table 
1  .1  iT  ?T^.  P'"*^  ^"  ^  ^^^^^^  ■  in  lOi  quarts  ?  in  2  gallons  i  in 
?>?^J..^'^^^"  '^  3^  P^'^ks  ?  in  3  quarters  1  ^  '    " 

24.  What  is  the  use  of  Wine  Measure?  19.  Repeat  the  Table 
-i>.  ±low  many  pipes  in  4  tunsi  Hogsheads  in  4  tunsi  Gallons  in 
^q;^arts^_m^quarts  1  Gills  in  20  pinis ?  in  10^  pints' 

poieJfo  ?a?v"rnSfd'iVe?enf  iJ?/''"'T'"..^'^°-^  •^"''''^  inches;  but  the  number  is  sup. 
has  receut]7p  onc^sed  m  ,hP  4L^i  ^-  ^"  ^"^"ectieut,  it  is  2198  cubic  inches.  CougreS 
has  not  yet£:rSnerJwSmtd''  ""'"""  standard  of  weights  and  measures,  but  is 


14  MENTAL   ARITHMETIC. 

ALE    OR     BEER    MEASURE,* 

86.  Is  used  for  measuring  malt  liquors. 

2  pints  [pt.] make  1  quart qt. 

4  quarts make  1  gallou gal. 

9  gallons •  make  1  firkin fir. 

2  firkins make  1  kilderkin.  .  .  .  kil. 

2  kilderkins make  1  barrel bl. 

36  gallons make  1  barrel bl. 

54  gallons make  1  hogshead.  .  .  .  hhd. 

2  hogsheads  ....*....  make  1  butt, bt. 

LONG    MEASURE, 

27.  For  measuring  length  without  regard  to  breadth  or  depth. 

1  barley  corn ;  3  barley  corns 1  inch. 

28.  Of  the  foregoing  lines  the  shorter  one  is  exactly  1  barley 
corn  in  length ;  then  3  times  the  length  of  this  line  makes  3  barley 
corns,  or  1  inch,  which  is  the  exact  length  of  the  longer  line.  In 
like  manner,  12  times  the  length  of  I  inch  makes  1  foot ;  3  times  the 
length  of  1  foot,  1  yard,  &c. 

3  barley  corns  [b.  c] make  1  inch in. 

12  inches make  1  foot ft. 

3  feet make  1  yard yd. 

5^  yards,  or  16|feet make  1  rod rd. 

40  rods,  or  220  yards make  1  furlong.  .  .  .  fur. 

8  furlongs,  or  1760  yards   .  .  make  1  mile m. 

3  miles make  1  league  ....  1. 

69|  statute  miles make  1  degree.  .  .  .  deg. 

60  geographical  miles make  1  degree.  .  .  .  deg. 

360  degrees  make  1  circle,  or  the  earth's  circumference. 

29.    DISTANCES DEPTHS HEIGHTS. 

4  inches  .  .  .  make  1  hand,  for  measuring  the  height  of  horses. 

6  points  .  .  .  make  1  line,  )  used  in  measuring  the  length  of  pen 
12  lines  ....  make  1  inch,  \      dulums  for  clocks. 

5  feet make  1  geometrical  space,  used  for  distances. 

6  feet make  1  fathom,  for  measuring  depths  at  sea. 

3  miles ....  make  1  league,  for  measuring  distances  at  sea. 

gunter's  chain. 

30:  For  measuring  distances,  and  the  length  or  breadth  of  land. 

7™  inches make  1  link. 

25  links make  1  pole. 

100  links make  1  chain. 

10  chains make  1  furlong. 

8  furlongs make  1  mile. 

1.  The  dry  gallon  contains  268^  cubic  inches ;  the  wine  gallon  231  cubic  inches 
and  the  beer  gallon  282  cubic  inches.  The  same  standards  continued  in  use  in  Great 
Britain,  as  late  as  1826,  when  the  act  of  Parliament  came  into  operation,  by  which  the 
Imperial  gallon  of  277y^'^'^  cubic  inches  was  substituted  for  the  dry,  beer  and  win© 
gallons. 


LAND,  OR  SQUARE  MEASURE 


15 


31:  The  chain  for  measuring  distances  varies  from  2  to  4  rods  in 
length,  reckoning  25  links  to  a  rod. 


QUE  STIC  NS. 

32.  Repeat  the  Table  of  Ale  Measure.  Its  use?  26.  Repeat  the 
Table  of  Long  Measure.  Its  use  1  27.  For  what  purpose  do  we 
use  the  league  and  fathom  1  the  geometrical  space,  lines  and  hands  T 

33.  How  many  barley  corns  in  20  inches'!  Inches  in  7  feet  6 
inches  1    Feet  in  I  Of  yards  ]    Yards  in  2  rods  1 

34.  How  many  leagues  in  38  miles  1  Furlongs  in  8f  miles  1  Fath- 
oms in  75  feetl  Feet  in  100  geometrical  spaces'? 


LAND,  OR  SQUARE  MEASURE, 

35.  For  measuring  superficies,  that  is,  surfaces  or  things,  whose 
length  and  breadth  are  considered  without  regard  to  depth ;  as,  land, 
paving,  flooring,  plastering,  roofing,  slating,  tiling,  &c. 


Fig. 


FiK.  2. 


Fig.  3. 


7 


36.  A  Square^  has  four  equal  sides,  and  four  equal  and  square 
corners,  commonly  called  Angles;  consequently  its  length  and 
breadth  are  equal  ;  as,  Fig.  1. 

37.  A  Parallelogram*  has  only  its  opposite  sides  equal,  and  at 
least  its  opposite  angles,  but  may  have  all  its  angles  equal ;  conse- 
quently it  has  more  length  than  breadth  ;  as,  figures  2  and  3. 

38.  Here  are  several  small  squares,  each 
of  which  we  will  suppose  is  1  inch  long 
and  1  inch  wide,  which  make  1  inch 
square,  and  therefore  is  said  to  contain  by 
measure,  1  square  inch. 

39.  If  we  count  the  square  inches  in 
the  two  top  rows,  they  will  make  2  times 
12  or  24  square  inches  ;  3  rows  will 
make  3  times  12  or  36  square  inches ;  4 
rows,  4  times  12  or  48  square  inches ; 
and  so  on  through  the  12  rows,  which 
make  12  times  12,  or  144  square  inches. 
40.  The  same  figure  taken  as  a  whole,  is  1  foot  square,  for  12 

inches  are  equal  to  1  foot ;  then  the  whole  contains  1  square  foot ; 

therefore,  1  square  foot  is  equal  to  144  square  inches. 

1  Square,  ZF.quarre.2  A  form  like  Figure  1,  above;  an  area  or  the  open  surface 
with  houses  on  four  sides ;  a  rule  tor  measuring ;  a  square  body  of  troops  ;  a  squadron ; 
a  quarternion ;  equality  ;  rule  ;  conformity ;  accord. 

2  Parallelogram  is  so  called  from  parallelos,  G.  equally  distant,  bwA  gramma  Q 
a  letter ;  because  its  opposite  sides  are  parallel ;  that  is  equally  distant  from  each  other 
in  all  their  parts  ;  of  course  parallel  lines  would  not  meet  if  continued  ever  so  far.  la 
common  use,  this  word  is  applied  to  any  quadrilateral,  or  four-sided  figures  of  more 
lengtb  than  breadth. 


] 

Fig.  4 

Small  Square!! 

. 

1 

16 


MENTAL   ARITHMETIC. 


Fig.  5. 

A  Parallelogram, 

Fig. 

6.  Squares. 

41.  This  figure  we  will  suppose  to  be  3 
feet  long  and  1  foot  wide ;  then  it  will  have  3 
squares  each  1  foot  square,  and  will  contain 
3  square  feet. 

42.  Three  rows  of  3  square  feet  in  each  row, 
will  contain  3  times  3,  or  9  square  feet,  but 
the  whole  figure  will  be  only  3  feet  square. 
Again,  as  3  feet  are  equal  to  1  yard ;  then  9 
square  feet  are  equal  to  1  square  yard. 

43.  Hence  we  see  that  3  square  feet  is  only 
one  third  as  much  as  3  feet  square,  making  a 
difference  of  6  square  feet. 

44.  Again  ;  suppose  Figure  5  to  contain  3 
square  miles  of  land,  then  3  miles  square  of  land;  as,  Figure  6,  would 
be  3  times  3,  or  9  square  miles,  making  a  difference  of  6  square  miles. 

45  Hence  a  square  foot,  yard,  &c.  may  be  of  any  shape  whatever, 
provided  the  foot  contains  exactly  144  squares,  each  1  inch  square, 
and  the  yard  9  squares,  each  1  foot  square,  &c. 

46.  From  the  above  we  learn  that  multiplying  the  length  of  any 
square  or  farallelogram  hy  its  breadth,  gives  its  square  measure,  or, 
as  it  is  sometimes  called,  its  square  contents,  as  in  the  following  Table  : 

144  square  inches  [sq.  in.]  .  make  1  square  foot sq.  ft. 

9  square  feet make  1  square  yard sq.  yd. 

30^  sq.  yd.  or  2721^  sq.  ft.  .  .  make  1  square  rod  or  pole.  .  .  sq.  r. 

40  square  rods make  1  rood R. 

4  roods make  1  acre A. 

640  square  acres make  1  square  mile sq.  m. 


QUESTIONS. 

47.  Repeat  the  Table  of  Land  or  Square  Measure.  Its  use  1  35. 
What  is  a  square  ■?  36.  Parallelogram  1  37.  What  is  meant  by  1 
square  foot  1    38.  39.  40.     What  by  1  square  yard  ^  41.  42. 

48.  What  is  the  difference  between  3  square  feet,  and  3  feet  square, 
and  why  1  42.  43— between  3  square  miles  and  3  miles  square  1  44. 

49.  How  many  square  feet  in  12  square  yards  1  square  rods  in  2^ 
roods  1  acres  in  38  roods  1 

50.  What  is  said  of  the  form  of  a  square  foot,  square  yard,  &c.  45. 
How  is  the  square  content  ascertained  1  46. 

51.  How  many  square  inches  then,  are  contained  in  a  small  slate 
4  inches  long  and  3  inches  wide  1  [3  times  4].  In  one  7  inches  long 
and  5  inches  wide  1  WTiat  is  the  form  of  such  slates,  and  why  1  37 

52.  How  many  square  feet  in  a  board  3  feet  long  and  1  foot  wide  ? 
[3].  In  one  7  feet  long  and  2  feet  widel  In  the  floor  of  a  room  10 
fp.et  long  and  7  feet  wide  1 


TIME.  17 

53.  How  many  square  yards  in  a  piece  of  carpeting  containing  54 
square  feet  1  Roods  in  a  piece  of  land  10  rods  long  and  4  rods  wide  1 
[1].    In  a  piece  10  rods  square  1 

SOLID    OR    CUBIC    MEASURE, 

54.  For  measuring  solids,  that  is,  things  that  have  three  dimen- 
sions, viz.,  length,  breadth,  and  depth  or  thickness ;  as  wood,  timber, 
stones,  masonry,  &c. 

55.  A  Cube  is  a  solid,  whose  length,  breadth,  and  thickness  are  all 
equal.     [See  the  Cubical  block.] 

56.  Thus  a  small  block  1  inch  long,  I  inch  wide,  and  1  inch  thick, 
is  called  a  cube,  and  is  said  to  contain  1  cubic  or  solid  inch.  Such  a 
block  has  of  course  six  equal  sides 

57.  Now  suppose  a  box  to  be  12  inches  square  in  the  inside ;  we 
can  then  place  on  the  bottom  12  rows  of  12  cubes,  each  containing  1 
solid  inch,  mj.king  in  all  144  cubic  blocks. 

58.  If  we  lay  another  tier  of  12  times  12,  or  144  similar  blocks 
upon  the  others,  we  shall  have  laid  2  times  144,  or  288  blocks,  and 
thus  we  might  continue  to  do,  till  we  had  laid  12  tiers,  which  would 
make  in  all  12  times  144,  or  1728  cubes ;  that  is,  so  many  solid 
inches,  and  all  in  the  form  of  a  perfect  cube,  for  the  whole  pile  would 
be  12  inches  long,  12  inches  wide,  and  12  inches  thick. 

59.  Then  as  12  inches  are  equal  to  1  foot,  the  above  cube  would 
be  1  foot  long,  1  foot  wide,  and  1  foot  thick,  which  makes  1  solid  or 
cubic  foot ;  hence  it  takes  1728  solid  or  cubic  inches  to  make  1  solid 
or  cubic  foot. 

60.  Hence  it  appears  that  a  solid  or  cubic  inch,  foot,  difc,  arise 
from  multiplying  the  length  of  a  solid  by  its  breadth,  and  that  result 
by  its  thickness,  as  in  the  following  Table  : 

1728  cubic  inches  [c.  in.]  ....  make  1  cubic  foot.  .  .  eft. 

27  cubic  feet make  1  cubic  yard.  .  .  c.  yd. 

50  cubic  feet  of  round  timber  .  make  1  ton T. 

40  cubic  feet  of  hewn  timber  .  make  1  ton T. 

42  cubic  feet  of  shipping  .  .  .  make  1  ton T. 

16  cubic  feet make  1  cord  foot.    .  .  c.  ft. 

8  cord  feet,  or  128  cubic  feet  make  1  cord  of  wood.  C. 

QUESTIONS. 

61.  How  many  dimensions  has  a  cube?  See  54. 55.  How  many  sides 
has  it  1    See  56.    What  is  meant  by  one  solid  or  cubic  inch  ?    See  56. 

62.  How  many  such  blocks  will  exactly  cover  the  space  of  12 
inches  square  1  See  57.  How  many  such  blocks  would  2  tiers  re- 
quire T    How  many  would  12  tiers  require  ?  See  58. 

63.  What  would  be  the  proper  name  for  the  form  of  such  a  pile  of 
blocks  when  taken  as  a  whole,  and  why  1  58.  What  is  meant  by  1 
cubic  foot,  59. 

64.  How  is  the  solid  content  of  a  cube  obtained  1  60.  How  many 
solid  feet  in  a  cubic  block  2  feet  long,  and  2  feet  wide,  and  2  feet 
thick  1  [8]    In  one  10  feet  long,  10  feet  thick,  and  10  feet  wide  \ 


A 8  MENTAL    ARITHMETIC. 

64.  How  many  cord  feet  in  32  solid  feet  ?  in  64  solid  feet  1  How 
many  cubic  feet  in  ^  of  a  cord  of  wood  ]  in  1  cord  ?  Repeat  th» 
Table  of  Cubic  Measure.    Its  use  ?  54. 

TIME. 

65.  Which  is  reckoned  by  years,  months,  days,  &c. 

60  seconds  [sec] make  1  minute.  .  .  m. 

60  minutes make  1  hour.    .  .  .  h. 

24  hours make  1  day d. 

365  days make  1  year.    .  .  .  Y. 

7  days  . make  1  week. .  .  .  w. 

4  weeks  [in  common  reckoning,]  .  .  .  make  1  month.    .  .  mo. 

52  weeks  [in  common  reckoning,]  .  .  .  make  1  year.    .  .  .  Y. 

30  days  [in  common  reckoning,]  .  "  .  .  make  1  month.  .  .  mo. 

12  months make  1  year.   .  .  .  Y. 

100  years make  1  century.    .  C. 

66.  The  number  of  days  in  each  month  are  as  follows  : — 
January  31  days.  May  31  days,  September  30  days, 
February  28  days,           June  30  days,            October  31  days, 
March  31  days,               July  31  days,  November  30  days, 
April  30  days,                 August  31  days,         December  31  days 

67.  The  days  in  each  month  are  often  expressed  thus  : 
Thirty  days  has  September,  April,  June  and  November ; 
February  has  twenty-eight,  and  thirty-one  the  others  rate. 
Except  inleapyear,'  happening  once  in  four. 

When  we  give  to  February  one  day  more. 

68.  A  naturaP  day^  is  .  .  24  hours. 

A  Lunar*  month'^    ....     4  weeks  or  28  days. 

A  Solar^  year'' 365  days,  5h.  48m.  48  sec.  [nearly]. 

A  Civil^  year' 12  calender^"  months^^  or  365  days. 

A  Julian^^  year^^  ....     13  lunar  months.  Id.  6 h.  or 365^  days. 

69.  When  any  year  can  be  divided  by  4  without  a  remainder,  it  is 
leapyear,^  except  the  centuriaP  years,  which  are  explained  below. 

1  Leapyear.    So  called,  because  the  year  leaps  as  it  were  from  365  to  .%6  days. 

2  Centurial,  [L.  centuria,  a  century. '\  Of,  or  belonging  to  a  century. 

3  A  Natural  Day  is  the  period  in  which  the  earth  revolves  (13)  on  its  axis,  (14) 
being  once  in  every  24  hours. 

4  Lunar,  ih.lunaMe  mcon.']  Pertaining  to  the  moon. 

5  A  Lunar  Month,  is  the  period  of  time  required  for  the  revolution  (13)  of  the  moon 
round  the  earth,  being,  strictly  speaking,  27  days,  7  hours,  43  minutes,  5  seconds. 

6  Solar,  \\a.  sol,the  sun.l  Of,  or  belonging  to  the  sun. 

7  A  Solar  Year,  or  A  Year,  properly  speaking,  is  the  period  which  the  earth  re- 
volves (13)  round  the  sun,  occurring  once  in  every  365  days,  5  hours,  48  minutes  and  a 
trifle  more  than  48  seconds.  This  period  of  revolution  is  therefore  called  the  Solar  (6) 
or  Natural  Year. 

8  Civil,  [L.  cwis.]  Relating  to  the  community  or  government ;  polite. 

9  A  Civil  Year,  is  the  period  of  time  established  by  law. 

10  Calendar,  (L-  calendanum.Z  A  register  in  which  the  montlis  and  days  are  set 
down  in  order ;  an  almanac. 

11  A  Calendar  Month,  is  a  solar  month,  as  it  stands  in  almanacs. 

12  A  .Julian  Year,  is  so  called  from  Julius  Caesar,  Emperor  of  Rome,  who  from  a 
desire  to  make  the  civil  year  correspond  (15)  with  the  solar,  ordered  it  to  consist  of  13 
months  1  day  and  6  hours,  or  365i  days ;  and  in  common  reckoning  only  365  days. 
Dropping  the  6  hours,  or  i  of  a  day  for  4  years,  makes  a  loss  of  one  day,  which  is 


CIRCULAR    MOTION.  19 

70.  When  there  is  a  remainder,  it  denotes  the  number  of  years 
since  the  last  leapyear ;  as,  1836,  1840,  1844,  but  in  1839,  for  in- 
stance, there  will  be  3  over;  1839,  then,  is  the  3d  year  since  the 
last  leapyear. 

71.  How  is  Time  reckoned?  65.  Repeat  the  Table,  65.  How 
many  days  [See  66]  has  January  1  February  1  March  1  April  ] 
June'?  August]  December] 

72.  How  are  the  days  in  each  month  expressed  ]  67.  How  may 
a  leapyear  be  known  ]   69 . 

73.  How  many  minutes  in  93  seconds?  in  1  hour  and  37  minutes'? 
days  in  4,  weeks?  years  in  60  weeks?  months  in  70  days?  in  1  cen- 
tury ? 


CIRCULAR    MOTION. 

74.  For  calculating  the  motions  of  the  planets,^  and  computing 
latitude^  and  longitude.^ 

60  seconds  (^^) make  1  minute.  .  .  .  '' 

60  minutes make  1  degree.  .  .  .  o 

30  degrees make  1  sign s. 

12  signs  or  360  degrees,  the  whole  circle  of  the  Zodiac.^ 

75.  Every  circle,  whether  the  greatest  or  least  possible,  is  divided 
into  300  equal  parts,  called  degrees,  and  each  degree  into  60  equal 

1  Planet,  from  planao,  G.  to  wander,  means  a  celestial  (4)  body,  which  revolves 
about  the  sun  or  other  centre,  or  a  body  revolving  about  another  planet  as  its  centre. 

2  Latitude,  from  latus,  L.  side,  means  the  distance  North  or  South  of  the  equa- 
tor; (5)  breadth;  room;  space;  extent  of  meaning  ;  freedom  from  rules. 

3  Longitude,  from  longus,  L.  long,  means  in  its  application,  the  distance  of  any 
place  on  the  globe  from  another  place  eastward  or  westward ;  the  distance  of  any  place 
from  a  given  meridian. (6) 

4  Celestial,  [L.  coelestis.'i  Heavenly ;  belongingto  the  upper  regions,  or  that  a  part 
of  which  we  see  over  our  heads  hi  a  clear  sky. 

5  Eq  uator,  L.  is  merely  an  imaginary  (8)  line  or  circle,  which  is  supposed  to  pass 
quite  round  the  earth  from  East  to  West. 

6  Meridian,  from  mcridies,  L.  mid  day,  is  an  imaginary  circle,  which  is.  supposed  to 
pass  from  North  to  South  quite  round  the  earth  and  through  each  pole. 

7  Zodiac,  [L.  zodiacus.'i  A  girdle;  a  broad  circle  in  the  heavens,  containing  the  12 
signs  through  which  the  sun  passes  in  its  annual  (9)  revolution. 

8  Imaoinary,    Not  real ;  not  existing  at  all  in  fact. 

9  Annual,  from  annus,  L.  a  year,  means  yearly;  that  returns  every  year. 

made  up  by  adding,  every  fourth  year,  one  more  day,  making  366  days  to  the  year,  called 
Bissextile  (16)  or  Leapyear.  Bissextile  or  Leapyear  then  has  366  days,  and  happens 
once  in  4  years,  so  that  any  year  that  can  be  divided  by  4  without  a  remainder,  is 
called  Leapyear,  as  1836,  1840,  1844,  «fec.  with  the  following  exception,  viz :  By  this 
reckoning  tlie  civil  year  would  have  11  minutes  and  about  12  seconds  more  than  the  true 
solar  year ;  and  every  cent  u  rial  year,  that  is,  the  year  that  completes  a  century  would 
be  a  leapyear,  therefore  to  make  the  two  years  correspond,  it  was  ordered  that  3  centu- 
rial  years  in  succession  should  be  reckoned  common  years,  and  the  fourth  one  only,  a 
leapyear. 

13  Revolve,  [L.  reuoZt'o.]  To  turn  in  a  circle ;  to  roll  any  thing;  to  consider. 

14  Axis,  it,,  axis.']  Something  passing  through  the  centre  of  any  thing  on  which  it 
turns ;  an  imaginary  line  running  from  North  to  South  through  the  centre  of  the  earth. 

15  Correspond,  [L.  correspondeo.]  To  answer  to ;  communicate  with. 

16  Bissextile,  So  called  from  the  Latin  ftis, /wicc,  and  sextilisiha  sixth;  because 
in  that  year  the  6th  of  the  kalends  of  March  was  repeated  twice. 


20  MENTAL    ARITHMETIC. 

parts,  called  minutes,  and  each  minute  into  60  equal  parts,  called 
seconds. 

76.  The  circumference^  of  the  earth  is  a  great  circle  of  360  de- 
grees. On  this  circle  every  minute  is  reckoned  a  mile,  60  of  which 
are  equal  to  1  degree  or  about  69|^  statute  or  common  miles. 


77.    TABLE    OF    PARTICULARS. 

12  things make  1  dozen. 

•   12  dozen make  1  gross. 

12  gross  or  144  dozen    .  make  1  great  gross. 

20  things make  1  score. 

5  score make  1  hundred. 

24  sheets  of  paper  .  .  .  make  1  quire. 

20  quires make  1  ream. 

200  pounds        make  1  barrel  of  pork. 

200  pounds make  1  barrel  of  beef. 

196  pounds make  1  barrel  of  flour. 

30  pounds make  1  barrel  of  anchovies. 

112  pounds make  1  barrel  of  raisins. 

256  pounds make  1  barrel  of  soap. 

200  pounds make  1  barrel  of  shad  or  salmon. 

7^  pounds make  1  gallon  of  train  oil. 

1 1  pounds make  1  gallon  of  molasses. 

14  pounds make  1  stone  of  iron  or  wood. 

8  pounds make  1  stone  of  meat. 

28  pounds make  1  tod. 

66  pounds make  1  firkin  of  butter. 

94  pounds make  1  firkin  of  soap. 

112  pounds make  1  quintal  offish 

364  pounds make  1  sack. 

19|^  cwt make  1  fother  of  lead. 

30  gallons make  1  barrel  offish. 

32  gallons make  1  barrel  of  cider. 

32  gallons make  1  barrel  of  herring,  England. 

42  gallons make  1  bar.  salmon,  eels  England. 

7-|  bushels make  1  hhd.  on  shore. 

8  bushels  salt make  1  hhd.  at  sea. 


QUESTIONS. 

78.  How  many  single  things  in  5  dozen?  In  1  gross?  In  5  score  1 
sheets  of  paper  in  2|  quires  1  quires  in  5^  reams  1 

79.  How  many  pounds  in  2^  barrels  of  pork?   In  f  of  a  barrel  of 
shad?  In  2 1  barrels  of  shad? 

1  Circumference,  from  circum,  L.  round,  and  fero,  L.  to  bring.    The  distance 
round  the  outside. 


TABLE  OF  ALIQUOT  PARTS.  21 

80.  BOOKS. 

A  FoLio^  is  when  a  sheet  is  folded  in  two  leaves. 
A  Quarto^  or  4to.,  is  when  a  sheet  is  folded  in  four  leaves. 
An  Octavo^  or  8vo.,  is  when  a  sheet  is  folded  in  eight  leaves 
A  Duodecimo*  or  12mo.,  is  when  a  sheet  is  folded  in  12  leaves. 
An  18mo.^  is  when  a  sheet  is  folded  in  18  leaves. 


ALIQUOT    PARTS. 

81.  An  Aliquot  Part  is  that  number  which  is  contained  in  an- 
other an  exact  number  of  times :  thus  5  is  an  aliquot  part  of  15,  for 
it  is  contained  in  15  exactly  3  times ;  that  is,  5  is  ^  of  15. 

82.  So  50  cents  and  25  cents  are  both  aUquot  parts  of  a  dollar; 
for  50  cents  =  (these  2  lines  mean  equal  to)  ^  of  100  cents  or  1  dol- 
lar, and  25  cents  =  4  of  1  dollar. 

83.  Again ;  every  6|-  cent  piece  is  an  aliquot  part  of  100  cents  or  1 
dollar,  being  exactly  y'^  of  a  dollar,  for  65^  times  16=  100. 

84.  Again;  every  12|  cent  piece  is  an  aliquot  part  of  1  dollar, 
being  exactly  \  of  a  dollar,  for  12|  times  8  =  100. 

85.  If  one  h\  cent  piece  is  yV?  then  two  6^  cent  pieces,  which 
make  12^  cents,  are  y^,  and  y^  are  equal  to  \;  for  8  times  \  are  |, 
which  are  equal  to  1  whole ;  and  8  times  y^  are  -J-f ,  which  are  also 
equal  to  1  whole. 


86.    TABLE    OF   ALIQUOT    PARTS. 

One 6|- cent  piece,  or   Cy  cents ^y'g- of  a  dollar. 

Two 6$-  cent  pieces,  or  12|-  cents  =  |  of  a  dollar. 

Three  .  .  .  .  6|  cent  pieces,  or  18^  cents=y^  of  a  dollar. 

Four    ....  6|- cent  pieces,  or   25  cents  ==  |  of  a  dollar. 

Five 6r  cent  pieces,  or  31^  cents=y%  of  a  dollar. 

Six 6*  cent  pieces,  or  37-^- cents  =  |  of  a  dollar. 

Seven  .  .  .  .  6|-  cent  pieces,  or  43|  cents  =y'^  of  a  dollar. 

Eight  ....  6  f  cent  pieces,  or    50  cents  ==  4'  of  a  dollar. 

Nine    ....  6^  cent  pieces,  or  56}  cents =-,^^  of  a  dollar. 

Ten Q\  cent  pieces,  or  62|  cents  =  |  of  a  dollar. 

Eleven    .  .  .  65- cent  pieces,  or  68?  cents  =||- of  a  dollar. 

Twelve  .  .  .  6}  cent  pieces,  or    75  cents— |  of  a  dollar. 

Thirteen    .  .  6}  cent  pieces,  or  81  f  cents =y|  of  a  dollar. 

Fourteen    .  .  Q\  cent  pieces,  or  87|  cents  =  1  of  a  dollar. 

Fifteen    ...  6}  cent  pieces,  or  93  J  cents=jf  of  a  dollar. 

Sixteen  .  .  .  6|  cent  pieces,  or  100  ceiits=ONE  Dollar. 

1  Folio,  from  the  Latin  folium  a  leaf. 

2  Quarto,  from  the  Latin  guarius,  \hefourt'i. 

3  Octavo,  from  the  Latin  octavvs.  the  eighth. 

4  Duodecimo.  Latin  duodecimus,  the  txoelfth. 

5  An  18mo.  from  cctavus,  L.  eighth,  and  dccimus,  L.  tenth,  both  making  eighteen. 


22  MENTAL    ARITHMETIC. 

One    ....  12 1  cent  piece,    or  121  cents  =|  of  a  dollar. 

Two  ....  121  cent  pieces,  or   25  cents  =  |-  of  a  dollar. 

Three   .  .  .  12^  cent  pieces,  or  37^  cents  =  |  of  a  dollar. 

Four  ....  12^-  cent  pieces,  or    50  cents  =  |  of  a  dollar. 

Five  ....  12^- cent  pieces,  or  62^  cents  =  I  of  a  dollar. 

Six,  ....  12|  cent  pieces,  or    75  cents  =  f  of  a  dollar. 

Seven  .  .  .  12l  cent  pieces,  or  87|  cents  =  |  of  a  dollar. 

Eight    ...  12^  cent  pieces,  or  100  cents = One  Dollar. 


QUESTIONS. 

87.  What  is  meant  by  an  aliquot  part?  81.  Why  is  5  an  aliquot 
part  of  15?  81. 

88.  What  aliquot  part  of  a  dollar  is  65- cents  ?  is  50  cents?  is  25 
cents  ?   121  cents  ?  Why  is  each  an  aliquot  part  ? 

89.  Why  is  j\  equal  to  \%  See  85. 

90.  Why  is  y*^  or  |  equal  to|?  A.  y^  is  equal  to  5^,  because  it 
means  4  of  16  equal  parts,  which  is  |  of  them :  and  |  is  equal  to  \, 
because  it  means  2  of  8  equal  parts,  which  is  also  5-  of  them. 

91.  How  many  cents  in  y^^  of  a  dollar  ?  in  |  ?  in  y\  ?  in  §  ?  in  y^^ 
in  f  ?  in  ||-  ?  in  1  ?  in  |  ?     Repeat  the  Table. 

92.  What  will  4  knives  cost,  at  Q\  cents  apiece  ?  At  12|  cents 
apiece  ?  At  25  cents  ?  At  50  cents  ? 

93.  How  many  yards  of  cloth  may  be  bought  for  1  dollar,  at  50 
cents  a  yard?  At  25  cents  a  yard  ?   At  12|  cents  ?  At  6|-  cents  ? 

94.  How  many  yards  then  would  5  dollars  buy  at  50  cents  a  yard  ? 
At  25  cents  ?  At  12}  cents  ?  At  6^  cents  ? 

95.  Mary,  having  purchased  29  yards  of  ribbon  for  6|  cents  a  yard, 
gave  the  merchant  a  2  dollar  bill;  how  much  change  ought  she  to 
have  received  ? 


MISCELLANEOUS    QUESTIONS. 

Vin.  1.  A  man  bought  12  bushels  of  wheat  for  30  dollars,  and 
sold  9  bushels  for  25  dollars.  How  many  bushels  had  he  left,  and 
what  did  they  cost  him  ? 

2.  Suppose  a  stage  goes  120  miles  in  12  hours,  and  a  railroad  car 
3  times  as  fast,  how  far  does  each  go  in  an  hour  ?  How  far  would 
they  be  apart  in  2  hours  after  they  had  started  ? 

3.  How  many  times  12  in  27?  2  ^2  times  12  are  how  many? 
How  many  times  11  in  30  ?  2~  times  11  are  how  many  ? 

4.  How  many  inches  in  8  nails?  yards  in  10  rods?  cents  in  5 
dollars  ?  shillings  in  2  pounds  2  shillings  ? 

5.  When  a  coat  costs  2  pounds  2  shillings  in  London,  how  many 
guineas,  at  21  shillings  each,  will  pay  for  it  ? 

6.  If  1  quart  of  rum  is  enough  to  make  one  man  act  like  a  fool ; 
how  many  at  that  rate  may  be  made  fools  by  10  gallons  ? 


MISCELLANEOUS    QUESTIONS.  23 

7.  What  would  a  narrow  strip  of  board  only  4  inches  wide  and  3 
feet  or  36  inches  long  cost,  at  2  cents  a  square  foot  ]  How  many 
square  rods  make  1  acre]  [4 times  40.] 

8.  When  a  piece  of  land  is  40  rods  long,  how  wide  must  it  be  to 
make  1  acre  ?  A.  4  rods,  [for  40  in  160  =  4  times.]  How  wide  to 
make  2  acres  1  When  a  piece  is  80  rods  long,  how  wide  must  it  be 
to  make  1  acre  1 — to  make  2  acres  1 

9.  How  many  solid  feet  in  a  block  whose  three  dimensions,  viz 
length,  breadth  and  thickness,  are  each  2  inches'?  How  many  when 
each  dimension  is  3  inches  T  is  4  inches  ?  is  5  inches  T 

10.  How  many  cord  feet  in  a  small  pile  of  wood,  4  feet  long,  2 
feet  wide  and  2  feet  deep"?  How  many  in  a  pile  8  feet  long,  2  feet 
wide  and  2  feet  high  ]  In  one  8  feet  long,  4  feet  wide  and  4  feet  high  ? 

11.  If  a  man  earns  1  dollar  a  day,  how  much  will  he  earn  in  the 
lawful  days  for  labor  in  1  week  1  in  If  week  1  in  3f  weeks'? 

12.  At  1  dollar  a  dav,  how  many  weeks'  labor  may  be  hired  for 
13doUarsU2i]  for  22  dollars?  for  20  dollars  1  for  32 dollars ? 

13.  What  will  16  pounds  of  butter  cost  at  12|  cents  a  pound  1 
What  will  24  pounds  cost  1  25  pounds  cost  1 

14.  What  is  the  cost  of  4  bushels  of  potatoes  at  25  cents  a  bushel  1 
of  20  bushels  ?  40  bushels  1  41  bushels  1 

15.  When  3  bushels  of  dried  apples  sell  for  6  dollars,  what  are 
they  a  bushel  ?  What  would  2  bushels  cost  ?  8  cost  ?  20  cost  ? 

16.  If  5  yards  of  broadcloth  sell  for  20  dollars,  what  will  8  yards 
seU  for  at  the  same  rate '?  Find  the  price  of  1  yard  first.  What  would 
be  the  price  of  9  yards'?   11  yards'?   12  yards'?  20  yards'? 

17.  If  8  pounds  of  sugar  cost  1  dollar,  what  will  3  pounds  cosf? 
How  many  pounds  may  be  bought  for  4  dollars '?  for  5  doUars '?  for 
lOdoUars? 

18.  When  4  bushels  of  oats  cost  2  dollars,  how  many  bushels  may 
be  bought  for  2  dollars  50  cents '?  for  3  dollars  50  cents  1  for  5  dol- 
lars ?  for  8  dollars  50  cents  1 

19.  How  many  wholes  are  |,  f,  1  and  |?  y^^,  f_^  _^^  _s_  ^nd  y^^l 
I  f  and  |l  1  and  |  ]  [li]  ^  I  I  h  I  and  f  1 

20.  When  a  floor  is  4  feet  square,  how  many  square  feet  does  it 
contain'?  [4  times  4.] 

21.  What  then  is  the  square  of  4  ?  [16.]  What  the  square  of  5  ? 
[5  times  5.]  What  is  the  square  of  6 1  of  71  of  8?  of  10?  of  20? 

22.  The  4,  before  it  is  multiplied  by  itself,  is  called  the  square 
root  of  16  ;^  what  then  is  thp  square  root  of  25,  and  why  ? 

A.  5,  because  5  times  5  are  25 

23.  What  is  the  square  root  of  4  ?  of  9  ?  16  ?  of  36  ?  of  64  ?  of 
100?  of  144?  of  400?  of  10000. 

1  Root.  That  part  of  a  plant  which  penetrates  the  ground  and  supports  the  plant ; 
the  first  ancestors ;  the  original  cause  of  any  thing.  To  take  root  means  to  become 
firmly  fixed.  The  square  root  of  any  number  is  so  called  because  it  is  the  first  number 
that  is  repeated  or  multiplied. 


24  MENTAL    ARITHMETIC 

24.  In  a  square  room  which  is  calculated  to  accommodate  100 
boys,  how  many  must  sit  on  a  single  bench  1 

25.  Suppose  400  scholars  should  wish  to  form  themselves  into  a 
solid  phalanx/  or  square  body,  how  many  must  stand  in  each  rank'"* 
and  file  1=*     A.  20. 

26.  What  is  the  cubic  or  solid  content  of  a  regular  cube  10  inches 
long,  10  inches  wide,  and  10  inches  thick] 

27.  What  is  the  10  and  1000  each  called?  A.  The  10  is  called 
the  cube  root  of  the  1000,  and  the  1000  the  cube  of  10. 

28.  What  then  is  the  cube  of  2,  and  why?  A.  8,  because  2  times 
2  are  4,  and  2  times  4  are  8. 

29.  What  is  the  cube  of  3  ?     What  is  the  cube  of  4 1 

30.  What  is  the  cube  root  of  8,  and  why?  A.  2,  because  2 times 
2  are  4,  and  2  times  4  are  8. 

31.  What  is  the  cube  root  of  271  of  125?  of  1000? 

32.  What  is  the  length  of  each  side  of  a  cubical  block  which  con- 
tains 1000  solid  or  cubic  inches  ?     What  is  the  cube  of  10  ? 

33.  6  is  I  of  what  number  ?  10  is  ^  of  what  number  ? 

34.  12  is  I  of  what  number  ?   11  is  ^  of  what  number  ? 

35.  What  is  that  number  |  of  which  is  9  ?  |  of  which  is  8  ?  ^  of 
which  is  10  ?  yV  of  which  is  12  ?  j^  of  which  is  11  ? 

36.  What  number  is  that  f  of  which  is  10  ?  ^  must  be  5.     A.   15. 

37.  What  number  is  that  f  of  which  is  24  ?  Find  ^  first.     A.  32. 

38.  What  number  is  that  f^  of  which  is  16  ? — |  of  v/hich  is  30  ? 
1  of  which  is  63 1  -f  of  which  is  60  ?  f  of  which  is  72  ? 

39.  If  f  of  a  barrel  of  flour  cost  4  dollars ;  what  will  ^  of  a  bar- 
rel cost  ?  What  will  the  whole  barrel  cost  [|]  ? 

40.  A  man  bought  ^  of  a  load  of  hay  for  6  dollars,  what  was  the 
whole  load  worth  at  that  rate  ?  Find  the  value  of  }  first. 

41.  Henry's  age  is  14  years,  which  is  |  of  his  brother's  age. 
How  old  is  his  brother  ?     Find  how  much  ^  is  first. 

42.  A  man  lost  15  dollars,  which  was  y^  of  all  the  money  he  had ; 
how  much  had  he  ?  How  much  had  he  left  ? 

43.  A  man,  who  owed  a  certain  sum  of  money  paid  12  dollars, 
which  was  y  2  of  the  debt ;  how  much  remained  unpaid  ? 

44.  A  man,  who  lent  a  certain  sum  of  money,  could  collect  only  8 
dollars,  which  was  f  of  it ;  how  much  did  he  lose  ? 

45.  If  a  man,  having  a  quantity  of  flour  on  hand,  sells  20  barrels, 
which  is  f  of  it ;  how  much  will  he  have  left  ? 

46.  Suppose  a  man  sells  I  of  a  barrel  of  flour,  for  14  dollars,  what 
will  the  remainder  of  the  barrel  bring  at  that  rate  ? 

1  Phalanx.  A  square  battalion  or  body  of  soldiers,  formed  in  ranks  and  files  close 
and  deep ;  any  body  of  troops  or  men  formed  in  close  array,  or  any  combination  of  people 
distinguished  for  their  intrepidity  and  union. 

2  Kank.  a  row  or  line ;  men  standing  side  by  side  in  a  line ;  a  line  of  things ;  degree; 
class ;  order ;  degree  of  dignity. 

3  File.  A  thread,  string  or  line ;  a  bundle  of  papers  tied  together  with  the  title  to 
each  indorsed  ;  a  roll,  list,  or  catalogue ;  a  row  of  soldiers  ranged  one  behind  another 
from  front  to  rear. 


MISCELLANEOUS    QUESTIONS.  25 

47.  A  boy  having  a  stick,  broke  it  into  two  parts,  one  of  which 
was  2  feet  long,  or  ^  of  the  length  of  both  parts ;  what  was  the 
length  of  the  stick  before  it  was  broken  1 

48.  If  I  and  §  of  a  number  are  16  ;  what  is  that  number! 
Note — Say  |  and  §  are  f  which  is  16  ;  then  |  is  j  of  16,  which  is 

4,  and  f  is  5  times  4,  which  is  20,     A.  20. 

49.  If  f  and  f  of  a  number  are  10  ;  what  is  that  number  1 
60    If  |-  and  f  of  a  number  ar6  70  ;  what  is  that  number  1 

51.  There  is  a  pole  erected  so  that  ^  stands  in  the  mud,  f  in  the 
water,  and  the  rest  which  is  10  feet  above  water ;  what  is  the  entire 
length  of  the  pole  1 

62.  There  is  a  pole,  y  above  water,  f  in  the  water ;  and  8  feet  in 
the  mud ;  what  is  the  entire  length  of  the  pole  1 

53.  Four  men  A,  B,  C,  and  D  purchased  a  sloop  together.  A  took  | 
of  it,  B.  f ,  C.  I  and  D,  the  rest,  which  cost  him  100  dollars.  What 
was  D's  partT  What  did  the  other  parts  severally  cosf?  What  did 
the  whole  sloop  cost  1 

54.  The  fifth  part  of  an  army  was  killed ;  §  of  it  taken  prisoners, 
and  1000  fled ;  how  many  were  there  in  the  army  1  How  many  were 
killed  l  How  many  were  taken  prisoners  1 

65.  In  an  orchard  of  fruit  trees,  f  of  them  bear  apples,  f  bear 
plums  :  8  bear  peaches,  and  2  bear  cherries  :  how  many  trees  of  each 
sort  are  there  in  the  orchard  !  How  many  trees  does  the  orchard  con- 
tain? 8  trees  and  2  trees  are  10  trees  which  are  J. 

56.  In  a  certain  school  4  of  the  pupils  study  Arithmetic,  -f  study 
Grammar  and  10  only  read  and  spell :  what  is  the  number  of  scholars 
in  the  school?  What  is  the  number  in  Arithmetic?  What  is  the 
number  in  Grammar  ? 

57.  A  man  having  16  oranges  would  divide  them  so  that  his  own 
son  Samuel  may  have  4  more  than  his  neighbor's  son  George ;  How 
many  must  he  give  to  each  ? 

Note — Give  4  to  Samuel  first  then  divide  the  rest  equally  between 
the  two?  A.  George  6,  and  Samuel  10. 

68.  A  gentleman  bought  a  horse  and  carriage  for  240  dollars,  paying 
40  dollars  more  for  the  horse  than  for  the  carriage  ;  what  did  each 
cost? 

59.  A  man  and  a  boy  were  both  hired  for  20  dollars  a  month,  the 
man  receiving  4  dollars  a  month  more  than  the  boy ;  what  would  the 
wages  of  each  amount  to  in  a  year  ? 

60.  A  man,  woman,  and  boy  were  hired  a  week  for  21  dollars ;  the 
woman  to  receive  5  dollars  more  than  the  boy,  and  the  man  5  dollars 
more  than  the  woman ;  at  that  rate  what  would  the  wages  of  each 
amount  to  in  one  month. 


arithmetic; 


PART    SECOND. 

AS  CONSISTING  BOTH  IN  THEOKYS  AND  PRACTICE'. 

QUANTITY  AND  NUMBER. 

IX.  1.  Quantity*  is  any  thing  that  maybe  increased  or  dimin- 
ished ;  as,  a  sum  of  money,  a  line,  weight. 

2.  A  Quantity  is  ascertained  to  be  great  or  small,  much  or  little, 
only  in  comparison  with  a  known  quantity  of  the  same  kind,  which  is 
either  greater  or  smaller. 

3.  For  example,  ten  thousand  hogsheads  of  water  is  a  great 
quantity,  compared  with  one  gill  of  water,  but  quite  a  small  quantity, 
compared  with  the  water  in  the  ocean. 

4.  A  Unit,*  which  represents^  a  single  thing ;  as,  1  hat,  1  ounce, 
&c.  is  fixed  upon  as  the  criterion''  or  known  quantity  by  which  to 
measure  all  other  quantities  of  that  kind. 

5.  Thus  2  would  express  a  quantity  2  times  as  great  as  1,  that  is, 
2  units ;  3,  3  times  as  great,  or  3  units,  and  so  on. 

6.  Quantities  then  of  every  kind  are  properly  expressed  by  Num- 
bers ;  as  5  bushels  of  rye,  5  oranges,  &c. 

7.  A  Concrete^  Number  has  reference  to  some  particular  object 
or  objects ;  as,  1  man,  2  dollars,  3  benches, 

8.  An  Abstract^  Number  has  no  reference  to  any  object  what- 
ever; as  1,  2,  3. 

IX.  Q.  What  is  Quantity?  1.  How  is  a  quantity  ascertained  to  be  great  or 
small  ?  2.  Give  an  example  ?  3.  What  is  the  criterion  for  estimating  different 
quantities?  4.  Illustrate  it?  5.  How  are  quantities  expressed?  6.  What  is  a 
Concrete  Number?  7.  An  Abstract  Number  ?  8. 

1  Arithmetic,  IG.  Arithmetike'.']  Reckoning  by  numbers ;  calculating. 

2  Theory,  [F.  theorie.  L.  theoria.}  Speculation ;  a  system,  plan,  scheme ;  opposed 
to  practice. 

3  Practice,  [F.  pratique.^  Habit,  use,  dexterity,  method. 

4  Quantity,  lQuantitas.:i  Anything  that  may  be  increased  or  diminished;  bigness; 
bulk,  weight ;  measure. 

5  Unit,  [L.  unus.']  One;  a  word  denoting  a  single  thing. 

6  Represent.  To  show;  to  exhibit;  to  describe. 

7  Criterion,  [G.  kriterion.l  A  standard  of  judging ;  a  distinguishing  mark. 

8  Concrete,  [h.concretus.:i  United  in  one  mass ;  a  compound ;  a  term  involving  both 
the  thing  and  its  quality;  as,  a  white  fence ;  2  mellons ;  1  cent. 

9  Abstract,  [L.  abstractus.}  Separate  ;  distinct ;  expressing  only  quality  or  nuro 
ber ;  as,  whiteness ;  1,2,  3,  &c. 


ARITHMEtlC.  27 

9.  Denomination^  is  a  name  given  to  units  or  things  of  the  same 
sort  or  class ;  as,  4  dollars,  6  oranges,  10  pigeons. 

10.  A  Simple  Number  is  composed  of  units  of  the  same  value  or 
denomination. 

11.  A  Compound^  Number  is  composed  of  two  or  more  simple 
numbers  of  different  denominations,  but  of  the  same  genus,^  kind,  or 
general  class. 

12.  Thus  2  pounds  is  a  simple  number,  so  is  5  shillings  ;  but  2 
pounds  5  shillings  taken  together,  forms  a  compound  number,  for  it 
has  one  denomination  of  pounds,  another  of  shillings ;  but  both  are  of 
the  same  kind,  or  general  class,  viz.  money. 

13.  Arithmetic  treats*  of  numbers :  as  a  Science,^  it  explains 
their  properties ;  and  as  an  Art,"'  it  teaches  the  method  of  com- 
puting'* by  them. 

14.  Arithmetic  has  five  principal  rules  for  its  operation,  viz. 
Numeration,  Addition,  Substraction,  Multiplication,  and  Division, 
which  are  often  called  the  fundamental^  or  ground  rides  of  Arith- 
metic, because  they  are  the  foundation  of  all  the  other  rules.^" 

Q.  What  is  meant  by  denomination  ?  9.  What  is  a  Simple  Number  ?  10.  Com- 
pound Number?  11.  Give  an  example  of  a  compound  number.  Sec  12.  What 
IS  Arithmetic?  13.  When  is  it  regarded  as  a  Science?  13.  When  as  an  Art? 
13.  How  many  rules  has  it  for  its  operations  ?  14.  What  general  name  have 
these  rules,  and  why  ?  14. 

1  Denomination,  [L.  denomino.'\  The  act  of  naming,  a  name ;  a  vocal  sound,  a 
class,  sort,  or  name  of  a  species. 

2  Compound.  Composed  of  tw^oormore  ingredients;  united  in  one. 

3  Genus,  [L.  genus.^  A  general  name  for  several  species  ;*  a  class  of  greater  extent 
than  species  ;  thus  animal  is  a  genus  ;  embracing  a  great  variety  of  species ;  as  man, 
horse,  beast,  bird,  reptiles,  &c. 

4  Species.  A  kind,  sort,  class  ;  a  subdivision  of  a  general  sum  called  a  genus,  thus, 
things  that  resemble  each  other  in  several  particulars  form  a  species ;  when  several 
species  are  compared  together  and  we  observe  several  particulars  common  to  the  whole, 
they  form  a  genus ;  a  species  then  is  one  class  of  a  genus. 

5  Treat,  [F.  traiter.'\  To  handle;  to  use  ;  to  discourse  on  ;  to  entertain  without  ex- 
pense ;  to  negociate  ;  to  manage  in  the  application  of  remedies. 

6  Science,  [L.  Scientia.']  Knowledge  ;  a  system  comprising  the  theory  and  reasons 
without  any  practical  application  ;  and  therefore  stands  opposed  to  Art. 

7  Art,  [L.  art.]  Human  skill;  a  system  of  rules;  skill ;  dexterity. 

8  Computing.  Counting ;  numbering  ;  reckoning,  estimating. 

9.  Fundamental,  {.L.  fundament,  seat.]  Relating  to  the  foundation  or  basis. 

10  Addition  alone  may  not  inappropriately  be  styled  the  sole  or  fundamental  rule  of 
Arithmetic,  for  all  the  other  rules  are  easily  resolvable  (11)  into  this  single  one. 

10  Thus  4  in  20,  5  times,  because  5  times  4  are  20.  Division  then  involves  the  prin- 
ciple of  Multiplication. 

10  Again,  4  times  5  are  20,  because  5  and  5  and  5  and  5,  that  is  5  added  4  times,  makes 
20.    Hence  Multiplication  may  be  performed  by  Addition. 

10  Subtraction  too  is  virtually  perlbrmed  by  Addition,  for  5  from  20  leaves  15  only 
because  15  and  5  are  20. 

11  Resolvable,  [L.  resolvo.}  That  may  be  reduced  to  first  principles. 


28 


NUMERATION. 


NUMERATION.* 

X.  1.  There  are  three  methods,  as  we  have  seen,  by  which  num- 
bers are  represented  ;  viz.  by  words,  by  single  letters,  and  by  charac- 
ters usually  termed ^^-wre^^  and  sometimes  digits.^ 

2.  In  the  method  by  letters  ;  which  is  called  the  Roman  method, 
because  the  Romans  invented^  it;  are  employed  seven  letters  onlv 
viz.,  I,  V,  X,  L,  C,  D,  M.  ^' 

3.  This  method  possesses  some  advantages  over  that  by  figures ; 
but  it  has  become  nearly  obsolete,^  being  confined  principally  to  the 
numbering  of  chapters,  hymns,  &c. 

4.  The  method  by  figures ;  which  is  called  the  Arabic  method, 
because  the  Arabs  invented  it ;  is,  taken  as  a  whole,  by  far  the  shortest 
and  best  method  ever  devised.^ 

5.  In  this  method  are  employed  ten  figures  only;  viz. 
1,2,3,4,5,6,7,8,9,0; 

one        two        three       four        five         six       seven     eight        nine      cipher 

6.  The  first  nine  figures,  which  have  each  an  absolute^  value,  are 
called  significant^  Jigures,  to  distinguish  them  from  the  cipher;  which 
has  no  value  in  itself,  being  used  merely  to  fill  a  vacant^"  place.  The 
cipher  is  also  called  naught  or  zero.^ 

7.  By  variously  combining^^  these  ten  characters,  no  number  can 
be  conceived  of  too  great  to  be  represented  by  them,  as  will  appear 
by  the  sequel." 

X.Q.  How  are  numbers  represented?  1.  Describe  the  Roman  method  ?  2.  Is  it 
still  used  and  to  what  extent  ?  3.  Which  is  the  best  method  ?  4.  Describe  it?  4. 
What  characters  does  it  employ?  5.  Of  what  use  is  the  cipher?  6.  What  other 
names  has  it  ?  6.  What  names  have  the  other  characters  and  why  ?  6.  Can  a 
large  number  be  represented  by  so  few  characters  ?  7. 

1  Figures  were  introduced  into  Spain,  by  the  Arabs,  in  the  8th  century  (13)  and  from 
Spain  into  England  about  the  middle  of  the  11th  century  ;  most  eight  hundred  years  ago. 
On  the  continent  their  use  had  become  quite  extensive :  they  are  now  so  common,  tliat  if 
you  were  to  visit  China,  for  instance,  you  would  recognize  (14)  at  once  their  numerals,  (1.5) 
without  understanding  a  word  of  their  language. 

2  Digit,  [L.  digitus,  a  finger. 'i  The  measure  of  a  finger's  breadth,  or  the  fourth  of  an 
inch.     Figures  were  so  called  troni  counting  the  fingers  in  reckoning. 

3  The  character  0  is  called  a  cipher,  from  the  Arabic  word  tsphara,  which  signifies  a 
blank  or  void.  The  uses  of  this  character  in  numeration  are  so  important,  that  its  name 
cipher,  has  been  extended  to  the  whole  art  of  Arithmetic,  which  has  been  called  to  cipher, 
meaning  to  work  with  figures. 

4  Numeration,  [L.  numcratio.']  Numbering;  the  act  of  numbering, 

5  Invented,  ih.  inventus.']  Found  out;  devised;  contrived;  forged. 
C  Obsolete,  [L.  obsoletus.']  Gone  into  disuse ;  disused,  neglected. 

7  Devised,  [F.  deviser.]  Given  by  will;  bequeathed;  contrived  ;  invented. 

8  Absolute,  [L.  absolutus.]  Complete  ;  positive  ;  unconditional ;  independent 

9  Significant,  \L.  significans.]  Having  meaning  ;  expressive;  important. 

10  Vacant,  [F.fromL.vacans.]  Empty;  not  filled  ;  exhausted  of  air;  unoccupied. 

11  Combining,  [F.  combiner.]  Uniting  closely  ;  joining  ;  confederating  in  purpose. 

12  Sequel,  [F.  sequelle.]  That  which  follows  ;  consequence;  event. 

13  Century,  [L.  centuria.]  A  period  of  one  hundred  years. 

14  Recognize,    [L.  recogn'itio.]  To  know  again  ;  to  revise. 

15  Numerals,  [L.  numeratio.]  Characters  used  for  representing  numbers. 


NUMERATION.  29 

8.  The  Unit,  which  occupies  the  lowest  place  in  the  scale  of 
whole  numbers,  means  a  single  thing;  that  is,  one;  as,  1  hat,  1  boy. 

9.  The  Ten,  which  means  10  units,  is  the  least  number  that  is 
formed  by  the  union^  of  two  single  characters,  to  wit :  by  annexing' 
the  cipher  to  the  figure  1,  thus,  10 ;  twenty,  thus,  20  ;  thirty,  thus, 
30,  &c. 

10.  The  Hundred,  which  is  ten  times  10  units,  is  formed  by  an- 
nexing^ two  ciphers  to  the  figure  1,  thus,  100  ;  two  hundred,  thus, 
200  ;  three  hundred,  thus,  300,  &c. 

11.  The  Thousand,  which  means  ten  times  100  units,  is  formed 
by  annexing  three  ciphers  to  the  figure  1,  thus,  1000 ;  two  thousand, 
thus,  2000  ;  three  thousand,  thus,  3000,  &c. 

12.  The  Ten  Thousand,  which  is  ten  times  1000  units,  is  written 
thus,  10000  ;  one  hundred  thousand,  thus,  100000,  &c. 

13.  In  these  examples,  every  additional^  cipher  increases  the  value 
of  the  figure  1,  ten  times,  by  removing  it  one  place  further  towards 
the  left. 

14.  When  a  cipher  or  ciphers  occur*  on  the  extreme^  left  of  other 
figures,  they  possess  no  value,  as,  01 ;  or  001,  or  0001,  each  of 
which  examples  means  simply  1. 

15.  In  general,  the  removal  of  any  figure  one  place  further  towards 
the  left,  enhances^  its  value  ten  times. 

16.  Thus  in  1111,  the  first  figure  on  the  right  means  1  unit;  the 
next,  on  the  left  10  times  1  unit,  or  10  ;  the  next,  10  times  10  units, 
or  100  ;  the  next,  10  times  100  units,  or  1000 ;  all  making  one  thou- 
sand one  hundred  and  eleven. 

17.  Hence  numbers  increase  from  the  right  to  the  left  in  a  tenfold' 
proportion,^  as  in  the  following  Table, 

18.    NUMERATION      TABLE    1. 

10  units make  1  ten. 

10  tens make  1  hundred. 

10  hundreds make  1  thousand. 

10  thousands make  1  ten  thousand. 

10  ten  thousands make   1  hundred  thousand. 

10  hundred  thousands make  1  million. 

Q.  What  is  meant  by  the  unit  ?  8.  How  is  the  ten  formed  ?  9.  How,  the  hun 
dred  ?  10.  How,  the  thousand?  11.  How,  the  ten  thousand,  and  so  on?  12. 
What  effect  has  the  cipher  in  these  examples?  13.  What  does  the  figure  1  with 
either  one,  or  two,  or  three  ciphers  prefixed  represent?  14.  What  is  the  effect 
of  removing  a  figure  to  the  left?  15.  What  does  each  figure  1  in  1111  mean?  16 
What  is  the  law  of  increase  ?  17. 

1  Union,  [L.  unus,  one.'\  Forming  into  one  ;  bond  ;  affection ;  concord. 

2  Annexing,  [L.  annexus.}  Uniting  at  the  end ;  placing  after  something. 

3  Additional,  [L.  additio.l  That  which  is  added,  or  which  increases. 

4  Occur,  IL.  occurro.'\  Meet;  come  to  the  mind  ;  appear;  meet  the  eye. 

5  Extreme,  IL.  extremus,  the  last.]  Outermost;  falherest ;  most  pressing. 

6  Enhances,  Raises;  lifts;  advances;  increases;  heightens;  aggravates. 

7  Tenfold,  lien  and  fold.}  Ten  times  more  in  degree  or  extent. 

6  PROPaBTiON,  iL.  proportio.}  E^ual  degree  «r  equal  rate  J  symmetry. 

8* 


80  ARITHMETIC. 

19.  Suppose  a  curious  old  misdr  to  have  laid  up  several  bags  of 
dollars  containing  the  following  sums,  viz.  1  dollar,  10  dollars,  100 
dollars,  and  10000  dollars. 

20.  Then  1  bag  of  1  dollar  would  represent  1  unit ;  1  bag  of  10 
dollars,  1  ten ;  1  bag  of  100  dollars,  1  hundred;  1  bag  of  1000  dol- 
lars, 1  thousand  ;  and  1  bag  of  10000  dollars,  1  ten  thousand. 

21.  As  the  second  bag  and  all  the  succeeding  ones  are  each  but  a 
single  collection,  or  but  one  thing,  it  may  properly  be  called  a  unit,  as 
well  as  the  bag  that  contains  but  1  dollar. 

22.  Hence,  a  series,^  or  a  progressive^  order^  of  units  may  be 
established  in  which  each  succeeding*  one  shall  be  ten  times  the 
value  of  a  former  one, 

23.  Simple  units  may  then  be  denominated''  the  first  order,  tens, 
the  second  order ;  hundreds,  the  third  order,  and  so  on. 

24.  Thus  in  4689,  the  9  is  9  units  of  the  first  order ;  the  8,  8  units 
of  the  second  order ;  the  6,  6  units  of  the  third  order ;  the  4,  4  units 
of  the  fourth  order. 

25.  We  see  also  that  the  value  of  figures  depends  on  the  places 
they  occupy. 

26.  When  2  and  5,  for  instance,  stand  separately,  they  mean  simply 
2  units  and  5  units ;  but  placed  together,  they  may  mean  either  25 
units  or  52  units. 

27.  The  value  of  a  figure  standing  alone,  is  called  its  simple  valuer 
when  combined  with  other  figures,  its  local^  value. 

28.  To  express  two  thousand  three  hundred  and  forty-five,  we 
write  them  as  follows,  viz. 

w  to  Write  the  2  in  the  Thousands'  place ;  the  3  in  the 

^  §  Hundreds'  place  ;  the  4  in  the  Tens'  place  ;  and  the 

&  o  05  £      5  in  the  Units'  place.     This  is  called  Notation. 

a  §  H  S          29.  To  ascertain  if  we  have  correctly  written  the 

H  W  H  P      number,  begin  on  the  right  and  say ;  units,  tens,  hun- 

®  ^  ^  dreds,  thousands ;  then  begin  on  the  left  and  read, 

Q.  Repeat  the  Table  in  which  10  units  make  1  ten,  &c.  ?  18.  How  many  units 
are  there  in  2  tens  ?  in  5  tens?  tens  in  50  units  ?  in  100  units?  in  89  units?  [8  tens 
and  9  units.]  tens  in  95  units  ?  in  105  units?  [10  and  5  units.]  tens  in  165  units? 
hundreds,  tens  and  units  in  165  units  ?  [1  hun,  6  tens  and  8  units.]  in  456  units  ? 
units  in  4  hundreds  5  tens  and  6  units  1  What  is  meant  by  a  series  of  units  ?  22. 
Give  an  example  ?  20,  21.  What  constitute  the  several  orders  ?  23*  In  4689,  for 
instance,  point  out  the  different  orders  ?  24.  On  what  does  the  value  of  a  figure 
depend?  25.  In  expressing  2345,  by  figiires,  what  places  would  each  figure 
occupy  ?  28.  How  is  it  ascertained  if  it  be  correctly  written  ?  29.  What  num- 
ber will  1,  2,  and  3  represent,  taken  together  in  the  same  order  as  they  stand  ? 
A.  One  hundred  and  twenty-three.  What  number  y<fi\\  2,  3,  4,  and  5  represent, 
taken  in  like  manner  ? 

1  Series,  [L.  series.]  A  regular  succession  of  things  ;  course ;  order. 

2  Progressive.  Going  forward,  advancing  or  increa.sing  gradually. 

3  Order,  [L.  ordo.  F.  ordre.]  Method;  a  mandate;  rule,  rank,  class. 

4  Succeeding.  Following  in  order  ;  following  in  the  place  of  another. 
6  Denominated,  [L.  denominatus.]  Named;  called;  styled. 

6  Local,  [L.  loctu,  a  place.]  Of  or  belonging  to  a  place. 


NUMERATION. 


31 


givmg  to  each  figure  the  name  of  the  place  against  which  it  stands ; 
thus  2  thousand  3  hundred  and  45 ;  which  we  find  corresponds  with 
the  given  number.     This  is  called  Numeration. 

30.  Write  in  words  on  the  slate,  the  following  numbers : — 

10  10  0  10  0  0 

15  12  5  3  4  5  6 

25  521  6543 

89  891  8752 

31.  It  is  customary  to  separate  large  numbers  by  a  comma,  into 
parts  or  portions  called  periods  of  three  figures  each,  beginning  on  the 
right. 

32.  The  first  period ;  as  it  contains  units,  tens  of  units,  hundreds 
of  units,  is  called  the  period  of  Units. 

33.  The  next  left  hand  period,  for  a  similar  reason,  is  called  the 
period  of  Thousands,  and  so  on  as  in  the  following. 

34.    NUMERATION    TABLE    II. 

Period  of       Period  of  Period  of       Period  of 

BILLIONS.   MILLIONS.     THOUSANDS.     UNITS. 


i  : 

if 

VI 

1  0  0 

^Hundred  Millions.   .  . 

®Ten  Millions 

^Millions 

^Hundred  Thousands. . 
®Ten  Thousands.    .  .  . 

III 

,000 

,  read  100  billion 

•  2  0 

,000,000,000 

,  read  20  billion. 

.    •  3 

,000,000,000 

,  read  3  billion. 

•400,000,000 

,  read  400  million. 

c/\      nnn      ono 

,  read  50  million. 

,  read  6  million. 

,  read  700  thousand. 

,  read  80  thousand. 

,  read  9  thousand. 

,  read  9  hundred. 

,  read  ninety. 

,  read  one  hundred  twenty 

illion,  seven  hundred  eighty 

.    .    .  fi      0  0  0      0  0  0 

ly     A     A          AAA 

on       n  0  0 

.     .                            Q       0   ft   0 

.     .     .     .          .               .    Q   ft   ft 

9  0 

1  2  3 

three  billion 

,456,789,999 
four  hundred  fifty-six  m 

nine  thousan( 

i,  nine  hundred  and  ninety- 

nine. 

Q.  What  are  periods  of  figures  ?  31.  What  are  the  first,  second,  third,  «Scc,, 
periods  called  ?  32.  33.  Repeat  the  Numeration  Table  II ;  as,  units,  tens,  hun- 
dreds, &CC.,  as  far  as  hundred  billions  ?  34.  What  figures  in  the  Table  represent 
ninety?  Nine  hundred?  Nine  thousand?  Eighty  thousand?  Seven  hundred 
thousand?  Six  million?  Fifty  million  ?  Four  hundred  million  ?  Three  billion? 
Twenty  billion  ?  One  hundred  billion  ? 


8  ARITHMETIC. 

35.  Write  in  words  on  the  slate  the  following  numbers. 


36.  9  0,000 

37.  7  8,643 

38.  2  0  0,000 

39.  6  4  3,782 

40.  3,000,000 

41.  2,465,789 

42.  40,000,000 

43.  23,456,892 


44.  900,000,000 

45.  762,345,271 
40.     8,000,000,000 

47.  7,362,491,723 

48.  5  0,000,000,000 

49.  49,416,531,272 

50.  800,000,000,000 

51.  987,365,214,718 


52.  8,008,008,008  read  8  billion,  8    million,  &c. 

53.  70,070,070,070  read  70  billion,  70  million,  &c. 

54.  800, 800, 800, 800  read  800  bill'n,  800  mill'n,  &c. 

55.  6,000,600,000  read  6  bilFn,  and  600  thousand. 

56.  900, 000, 000,  009  read  900  biUion  and  9. 

57.  By  canceling^  all  the  ciphers  in  the  last  example  the  900  billion 
and  9  becomes  only  99. 

58.  Hence  be  careful  to  fill  all  vacant  plans  with  ciphers.* 

RECAPITULATION. 

59.  Notation  is  the  writing  of  numbers  in  figures ;  NumeratioNi 
the  reading  of  them  expressed  in  figures. 

RULE    FOR    WRITING    NUMBERS. 

60.  Begin  on  the  left  and  write  each  figure  according  to  its  value 
in  the  Numeration  Table,  taking  care  to  supply  all  vacant  places  with 
ciphers. 

Q.  What  caution  is  given  in  respect  to  vacant  places  ?  58.  Give  an  example 
of  its  importance  ?  57.  What  is  Notation  ?  59.  Numeration  ?  59.  What  is  the 
rule  for  writing  numbers  ?  60. 

*  The  practice  of  reckoning  only  three  figures  to  a  period,  is  derived  from  the  French, 
The  English  reckon  six  figures  for  a  period,  which  would  carry  the  millions'  place  in  the 
above  Table,  into  the  billions'  place.  One  billion  then,  by  the  French  mode,  expresses  a 
number  one  thousand  times  smaller  than  by  the  English  method ;  which,  as  you  may 
perceive,  greatly  diminishes  the  power  of  figures.  But  as  the  former  is  most  convenient, 
it  generally  has  the  preference. 

ENGLISH    METHOD    OF    NUMERATING. 


■^  a  w  ■"  .s  H  to  >-  .3  O  »l 


i-^llp  i^il.    KM.    i=il.    u 


_a-a      Tsi  Sra-^OK  ♦'S-a-^OM  ■"!0x)'^.2<"          ^^  m 

•a  g  CT3  03  d  'oSc'^ao  "^Scoisz  tsacxi^z  -o  a  g.^ 

£^52  =  5  £.§«2i^^  e2«5p=g  SS^^g^*  2  2  §  £ 

•oS^^So-S  .fe^g^i^  ^5S-gs2  ,SS§^S2  -feS^^.g 

»»J3S§!D  sg^agw  a§.a3g:5  s^SsgS  agSagS 
BHHKHO?  ffihHSHH  ffiHHKHW  KHE-ffiHS  ffiHHKhS 
333333,    333333,    333333,333333,    333333, 

Read  333  thousand  333  quadrillion,  333  thousand  333  trillion,  333  thousand  333  MlUon, 
333  thousand  333  million,  333  thousand  333. 

1  CAKoxbiirO}  [F.  canctlUr.}  Crossing ;  oblit«rating,  annulling. 


NUMERATION.  33 

PROOF,  OR  RULE  FOR  READING  NUMBERS. 

61.  Begin  on  the  right  and  numerate  by  saying  units,  tens,  hun- 
dreds, (Sfc. ;  then  begin  on  the  left  and  read,  joining  the  name  of  its 
place  to  each  figure;  which,  if  it  correspond  with  the  given  number, 
is  correctly  written. 

NUMERATION    TABLE    III. 

«       S       «       1       I       «/  -I 

?3^.  §g2  =3gs  li-.-gg-^l   .      §         .2         l-S 

05  g^3  |«^g  |C?-g  Sa^  SH^-^^pq  g2.S|n  H  §  §-^o 

6  55,5  55,55  5,555,5  5  5,5  55,555,555,555,5  5  5 

62.  Read  555  octillion,  555  septillion,  555  sextillion,  555  quintillion, 

655  quadrillion,  555  trillion,  555  billion,  555  million, 
555  thousand,  555. 

63.  Write  in  figures  on  the  slate,  the  following  numbers 

64.  Ninety-seven. 

65.  Four  hundred  and  twenty-five. 

66.  Three  thousand  and  five. 

67.  Forty-nine  thousand  five  hundred  and  twenty. 

68.  Six  hundred  and  fifty-two  thousand  five  hundred. 

69.  Eight  million  nine  hundred  and  forty  thousand. 

70.  One  hundred  and  one. 

71.  Five  thousand  and  five. 

72.  Four  thousand  two  hundred  and  eight. 

Q.  What  for  reading  numbers  ?  61.  Repeat  the  Numeration  Table  III.  How 
are  thirty  5s  in  succession  read  ?  How  would  thirty  3s  be  read  ? 

1  Primitive.  Original,  not  derived  from  any  thing ;  primary. 

2  Prefixing,  Uniting  at  the  beginning ;  placing  before. 

3  Numerals.  Of  or  belonging  to  number ;  consisting  of  numbers. 

4  Termination.  Limiting ;  bounding ;  ending ;  end  of  a  word. 

5  Modifications.  Changing  the  forms ;  altering  the  appearance. 

6  Euphony,  [G.  e«,  good,  and  phone,  sound.2  An  agreeable  sound. 

7  Prefix.  A  letter,  syllable,  or  word,  put  at  the  beginning  of  a  word. 

8  One,  two,  three,  and  up  to  twelve,  are  reckoned  primitive  (1)  words. 

9  Thirteen,  fourteen;  thee  and  ten,  four  and  ten,  &c. 

10  Twenty,  thirty,  &c.  are  derived  from  two  tens,  three  tens,  &o 

11  The  hundred  is  from  the  Latin  hun  or  hundred. 

12  The  Thousand  is  derived  from  the  Saxon  thousand.  This  and  the  two  preceding 
numerals  (3)  are  usually  considered  as  primitive  in  our  language. 

13  The  million  is  derived  from  the  French  million.  „    ■.    ,     • 

14  The  billion, trillion,  quadrillion,  &c.  are  formed  by  prefixing  (2)  the  Latm 
numerals  to  the  termination  (4;  illion,  with  such  slight  modifications  (5)  as  euphony  (6) 
requires.  The  Latin  prefixes  (7)  are  bis,  twice;  tres,  three;  quartuor,  for ;  qmnque, 
five  ;  sex,  six  ;  septem,  seven ;  octo,  eight ;  novem,  nine  ;  decern,  ten ;  undecim,  eleven  ; 
duodecim,  twelve ;  tredecim,  thirteen,  &c.  These  prefixes,  with  illion,  make  Bilhon, 
Trillion,  Quadrillion,  Quintillion,  Sextillion,  Septillion,  OctiUion,  NonUlton,  UndectUton, 
Duodecillion,  Tredecillion, 


m 


ARITHMETIC. 


73.  Three  hundred  thousand  five  hundred. 

74.  Six  million  one  hundred  thousand. 

75.  Four  million  four  thousand  and  forty-nine. 

76.  Seventeen  million  one  hundred  and  twenty-five. 

77.  One  billion,  one  million,  one  thousand  and  one. 

78.  Five  hundred  and  twenty-one  billion,  three  hundred  million, 
three  hundred  thousand  and  one. 

79.  Five  trillion,  five  billion,  five  million,  five  thousand  and  five 
hundred  and  fifty-five. 

80.  Six  quadrillion,  six  hundred  million,  four  hundred  and  fifty- 
nine  thousand  and  sixteen. 

81.  Two  hundred  and  fifty  quintillion,  six  quadrillion,  two  billion, 
three  hundred  and  forty  thousand,  t 

Figures  on  the  slate  are  vrritten  thus, — 

/  ^  J  ^  J  6  y  s  g  0 


SIMPLE   ADDITION/ 

XI.     1.  Add  together  9  dollars,  7  dollars,  5  dollars,  8  dollars,  6 
dollars,  4  dollars,  and  3  dollars,  thus  ; 


(1.) 

(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

9  dollars. 

8  tons. 

3  cents. 

8  ounces. 

5  mills. 

8  hats. 

7  dollars. 

5  tons. 

4  cents. 

9  ounces. 

8  mills. 

9  hats. 

5  dollars. 

6  tons. 

9  cents. 

7  ounces. 

3  mills. 

7  hats. 

8  dollars. 

9  tons. 

7  cents. 

8  ounces. 

9  mills. 

8  hats. 

6  dollars. 

7  tons. 

8  cents. 

5  ounces. 

6  mills. 

6  hats. 

4  dollars. 

3  tons. 

9  cents. 

9  ounces. 

7  mills. 

7  hats. 

3  dollars. 

4  tons. 

5  cents. 

6  ounces. 

8  mills. 

9  hats. 

*A.  42  dollars. 

* 

* 

* 

* 

* 

XI.  Q.  How  much  is  33  and  9?  What  is  the  42  called 

?  See  7. 

t  Remarks  to  the  Learner.  Aa  very  high  numbers  are  somewhat  difficult  to  appre- 
hend ;  it  may  not  be  amiss  to  illustrate,  by  a  few  examples  the  value  of  the  words  mil- 
lion, billion,  trillion,  and  quadrillion,  according  to  the  English  notation. 

Suppose  that  a  person  employed  in  telling  money,  reckons  a  hundred  pieces  in  a  minute, 
and  continues  to  do  so  twelve  hours  each  day,  he  will  take  nearly  fourteen  days  to  reckon 
a  million.    A  thousand  men  would  take  more  than  38  years  to  reckon  a  billion. 

The  inhabitants  of  the  United  States  in  1820,  were  about  10  million.  Now  if  we  sup- 
ix)se  all  these  persons  had  been  constantly  employed  in  counting  money  since  the  birth 
of  Christ,  they  could  not  as  yet  have  reckoned  a  trillion. 

Though  we  admit  the  earth,  from  the  creation,  to  have  been  as  populous  as  it  is  at 
present  (being  about  600  million,)  and  the  whole  human  race  to  have  been  counting 
money  without  intermission  ;  they  could  scarcely,  as  yet,  have  reckoned  the  five  hun- 
dredth part  of  a  quadrillion  of  pieces. 

1  Addition,  fL.  additio.]  Any  thing  added ;  adding;  joinuig ;  uniting  two  or  more 
numbers  in  one  sum. 

*(1).^.  42  dollars.  (2).  ^.  42tons.  (3.)  ^.  45  cents.  (4.)  A.  52 ounces.  {5.)  A. 
46  miUs.    (6.)  A.  54  hats. 


SIMPLE    ADDITION. 


85 


7.  The  Answer  in  adding  is  called  the  sum  or  amount. 

How  LT^'    °^"f  u  \''^''^'  ^^  ^"^  ^^"^«'  8  at  another,  9  at  another. 
How  many  cows  did  he  buy  in  all  t 


Proof.  Having  added  upwards  as  before ;  which 
makes  22,  draw  a  line  under  the  5  at  top;  then  add 
downwards  all  the  figures  under  the  5,  thus,  8  and 
9  are  17 ;  Then  if  the  17  and  the  5  at  top  make  22, 
as  they  do,  the  work  is  right. 


5  cows. 

8  cows. 

9  cows. 
A.  22  cows. 

17  cows. 
P.  22  cows. 

In  like  manner  perform  and  prove  the  following  examples 

(9-)  (10.)  (11.)  (12.) 

8  horses.       3  oxen.       9  calves.       8  shillings. 
8  calves.       9  shillings. 

7  calves. 
6  calves. 
5  calves. 
4  calves, 
3  calves. 
2  calves. 
9  calves. 

8  calves. 


8. 


9  horses 
7  horses. 

6  horses. 

7  horses. 

8  horses. 

4  horses. 

5  horses. 

6  horses. 
5  horses. 


2  oxen. 

5  oxen. 

6  oxen. 

8  oxen. 

9  oxen. 

7  oxen. 
4  oxen^ 
9  oxen. 

8  oxen. 


8  shillings. 

9  shillings. 

3  shillings. 
2  shillings. 

5  shillings. 

4  shillings. 

6  shillings. 

7  shillings. 


(13.) 
7  pence. 
5  pence. 
9  pence. 
4  pence. 

2  pence. 
4  pence. 

3  pence. 
2  pence. 
9  pence. 
6  pence. 


14.  Since  1  ten  and  1  unit,  for  instance,  make  neither  2  tens,  nor 
2  units,  (although  1  and  1  are  2) ;  but  1  ten,  which  is  10  units,  added 
to  1  unit  will  make  1 1  units,  thus  keeping  the  1  ten  in  its  proper  place, 
therefore  : — 

15.  Write  units  under  units,  tens  under  tens,  hundreds  under  hun- 
dreds, cfc,  then  add  each  column  separately. 


(16.) 
3  2  pints. 
3  5  pints. 
2  1  pints. 

1  pint. 


(17.) 

3  12  4 

4  1  1 

2  2  3 

1   1 


(18.) 
2 
5  2  4  3 

1  2  2 

2  1  1 


1  9 


(19.) 

2  2  2  2  2  2 

4  13  2  11 

2  4  3  4 

2  1 


20.  Ciphers  are  passed  over  in  adding,  because  they  are  used  to  M 
vacant  places.    See  x.  6. 
(21.)  (22.)  (23.)  (24) 

onn    ^2nn    510040  4003005005 
000     300    306054  420100 

103    2000         600  6001000201 

Q  0  4    4  0  0  1    9  0  0  3  0  0  5190002091 


25.  A  man  bought  a  farm  for  4,000  dollars ;  he  paid  for  his  cows 
405  dollars ;  for  his  horses  320  dollars ;  for  his  farming  utensils  60 

Q.  What  do  1  ten  and  1  unit  make  added  together?  See  14.  Why  not  2  tens  or 
2  umts  ?  How  then  should  units,  tens,  &c.,  be  written  and  added  ?  15.  How  are 
ciphers  to  be  regarded  m  adding?  20. 


3^ 


ARITHMETIC. 


dollars ;  and  his  expenses  for  securing  his  title  were   3   dollars. 
What  was  the  amount  of  the  whole  ]  A.  4,788  dollars. 

26.  Add  into  one  sum  forty,  two  hundred,  three  hundred  and  nine- 
teen, and  nine  hundred  and  forty.  A.   1499. 

27.  What  is  the  amount  of  one  thousand,  thirty-three  thousand 
three  hundred  and  sixty-one,  five  hundred  thousand  and  ten,  five  mil- 
lion and  five  thousand  ]  A.  5,539,371. 

28.  What  is  the  sum  of  five,  fifty,  five  hundred,  five  thousand,  fifty 
thousand,  five  hundred  thousand,  five  million,  fifty  million,  five  hun- 
dred million,  and  five  billion  %  A.  Ten  5s. 

29.  Add  together  8  trillion,  800  billion,  80  bilHon,  8  billion,  800 
million,  80  million,  8  million,  800  thousand,  80  thousand,  8  thousand, 
8  hundred,  80  and  8.  A.  Thirteen  8s. 

30.  Find  the  sum  of  3  trillion,  3  billion,  3  million,  3  thousand  and 
-3  ;  20  billion,  20  million,  20,020  ;  200  million ;  200  thousand  and  200 ; 
16  million,  16  thousand  and  16.  A.  3,023,239,239,239. 

31.  What  is  the  amount  of  4  million  and  six,  300  thousand  two 
hundred,  90  thousand  three  hundred  and  one,  4  thousand  two  hun- 
dred and  ten,  1  hundred  and  70,  eleven  and  11  A.  4,394,899. 

32.  A  cashier  has  in  one  drawer  5,305  dollars,  in  another  406  dol- 
lars, in  another  7,312,  in  another  2,309,  and  in  another  42.  What 
amount  has  he  in  aU  the  drawers  ? 

5  3  0  5         The  first  column  makes  24  units  or  2  tens  and  4 

4  0  6     units ;  write  down  only  the  4  units  and  add  in  the  2 

o  Q  n  Q     ^^"^  ^^^^  ^^^  column  of  tens,  which  is  called  carry- 

4  9     m^  1  for  every  ten.    The  2  to  carry  to  4  tens  makes 

6  and  1  are  7  tens,  and  none  to  carry.     The  next 

■^'  ^  ^  ^   '  ^     column  makes  13  hundreds,  or  1  thousand  and  3  hun- 
dreds ;  carry  the  1  thousand  to  the  thousands  ;  therefore  ; — 

33.  When  the  amount  of  any  single  column  is  10  or  more  : — Write 
down  only  the  right  hand  figure,  and  carry  the  left  hand  figure  or 
figures  to  the  next  column. 


(34.) 

3  9  0  6 

4  8  2  7 
3  3  3  9 
7  4  0  8 
9  5  12 

^.28992 

(35.) 
4  2  0  7 
9  3  5  3 

4  0  8 

7  8  19 
2  0  8 

(36.) 
4  12  8 

2  0  1 
7  0  9  5 
4  3  2  0 
9  0  9  0 

(37.) 

2  6  3  4 
2  0  9 

3  2  15 

8  1 
9 

38.  From  the  above  it  appears  that  we  begin  on  the  right  and  carry 
1  for  every  10 ;  because  figures  increase  from  the  right  to  the  left  in 
a  tenfold  proportion, 

Q.  When,  in  adding  several  columns,  the  right-hand  column  makes  24,  for 
instance ;  what  is  to  be  done  with  it  ?  32.  What  is  this  process  called  /  32. 
What  is  the  direction  for  carrying  ?  33.  What  is  the  reason  for  beginning  and 
carrying  in  this  manner?  38.  How  many  do  you  carry  when  the  sum  is  59  ?  115  ? 


SIMPLE    ADDITION.  37 

39.  Add  together  28,992  ;  21,995  ;  24,834  and  5,148.  A.  80,969. 

40.  A  man  bought  a  suit  of  clothes  for  57  dollars,  a  pair  of  boots 
for  8  dollars,  and  a  secretary  for  28  dollars.  What  is  the  amount  of 
the  whole  ^  A.  93  dollars. 

41.  In  an  orchard,  20  trees  bear  pears,  54  bear  peaches,  and  6  bear 
plums.     How  many  trees  are  there  in  the  orchard  1     A.  80  trees. 

42.  A  man  bought  a  barrel  of  flour  for  10  dollars,  a  barrel  of 
molasses  for  29  dollars,  and  a  barrel  of  pork  for  19  dollars.  What 
did  the  whole  cost  him  1  A.  58  dollars. 

43.  What  is  the  sum  of  eighty-seven,  two  hundred  and  seventeen, 
eight  thousand  nine  hundred  and  eighty-six,  and  nine  1    A.  9,299. 

44.  Find  the  sum  of  eight  thousand  and  twenty,  four  hundred  and 
seventy-nine,  thirty  thousand  and  sixty-five.  A.  38,564. 

45.  General  George  Washington  was  born  A.  D.^  1732,  and  lived 
67  years.     In  what  year  did  he  die  1  A.  1799. 

RECAPITULATION. 

46.  Addition  is  the  uniting  of  two  or  more  numbers  in  one  num- 
ber, which  is  called  their  sum  or  amount. 

47.  Simple  Addition  is  the  adding  of  numbers  of  the  same  denom- 
ination. 

RULE. 

48.  Write  the  nujnhers  in  columns,  so  that  units  may  be  added 
to  units,  tens  to  tens,  hundreds  to  hundreds,  <SfC. 

49.  Begin  on  the  right  and  place  underneath  each  column  its 
whole  amount,  unless  it  he  10  or  more  ;  in  which  case  set  down  the 
right-hand  figure  only,  and  carry  the  left'hand  one  to  the  next  column 

50.  Do  the  same  with  each  column  to  the  last,  under  which  write 
the  whole  sum. 

51.  Proof.  Omit  the  top  line,  and  find  the  sum  of  all  the  rest, 
adding  downwards  ;  if  the  numbers  were  before  added  upwards  and 
vice  versa,^  Then  if  this  amount  added  to  the  top  line,  corresponds 
with  the  first  amount,  the  work  is  supposed  to  be  right. 

52.  Or,  more  practically,  add  the  figures  downwards  without  omitting 
the  top  line,  and  if  the  two  amounts  agree  they  will  probably  be  right. 

53.  The  rule  as  well  as  the  proof  is  based^  on  the  well  known 
axiom,*  that  the  whole  is  equal  to  the  sum  of  all  its  parts. 

Q.  What  is  Addition?  46.  Simple  Addition?  47.  Rule?  48,  49,  50.  What 
are  the  two  methods  of  proof?  51.  52.  What  is  the  reason  for  both?  53. 

1  A.  D.  The  A.  stands  for  anno,  L.  for  year;  and  D.  for  Pomini,  L.  of  our  Lord.  Hence 
Anno  Domini  means,  in  the  year  of  our  Lord;  and  A.  D.  1732  means  so  many  years 
since  Christ,  or  our  Saviour  came  on  earth. 

2  Vice  versa.  That  is,  upwards  if  the  figures  were  before  added  downwards. 
Generally  vice  versa,  from  the  Latin,  means  the  terms  being  exchanged;  thus,  the  gen- 
erous should  be  rich  and  vice  versa,  that  is,  the  rich  should  be  generous. 

3  Based,  [L.  basis,  foundation.]  Founded;  reduced  in  value. 

4  Axiom,  [G.  axioma.i  A  self-evident  truth ;  that  which  is  so  plain,  that  no  proof  can 
make  it  any  plainer. 

4 


I 

ARITHMETIC. 

(54.) 

(55.) 

(56.) 

(57.) 

(58.) 

3  5 

3  1  3 

16  4  5 

13  2  13 

4  5  6  7  3  2 

6  4 

2  8  0 

3  2  1 

2  4  5  12 

12  12  12 

2  1 

7  4  1 

4  6  10 

5  2  10  8 

7  8  4  5  0  3 

1  8 

2  4  0 

5  3  8  0 

6  0  3  8  9 

9  0  8  7  6  2 

1  2 

3  9  1 

5  2  10 

7  8  9  7  8 

7  3  6  5  4  2 

59.  Add    into    one    sum    150;    1,965;  17,172;    229,200,    and 
3,007,751.  A.  3,256,238. 

60.  A  father  gave  to  his  oldest  son  4,200  dollars,  to  his  second 
2,300  dollars,  and  to  his  youngest  1,560  dollars.  What  is  the  amount 


of  these  several  sumsl 

A.  8,060  dollars. 

(61.) 

(62.) 

987654321 

87654210345965  2 

98765432 

1300000000 

9  8  7  6  5  4  3 

3210123456789 

9  8  7  6  5  4 

9  9  9  9 

9  8  7  6  5 

88885555 

9  8  7  6 

444444444444444 

9  8  7 

3  2  10  7  5  2  0 

9  8 

786743512 

9 

8  1  5 

1 

3  9  7  6  7 

63.  Find  the  sum  of  1  billion,  97  million,  393  thousand,  686,  and 
1  quadrillion  324  trillion,  198  billion,  879  million,  148  thousand,  and 
63.  A.   1,324,199,976,541,739. 

64.  A  gentleman  purchased  a  ship  for  25,000  dollars,  and  sold  it  for 
3,715  dollars  more  than  it  cost  him.     What  did  he  get  for  it  1 

A.  28,715  dollars. 

65.  A  gentleman  sold  a  tract  of  wild  land  for  13,000  dollars, 
which  was  1,750  dollars  less  than  it  cost  him.  What  did  he  give 
forit^  A.   14,750  dollars. 

66.  A  merchant  had  a  store-house  in  which  he  had  at  one  time 
6,000  bushels  of  corn,  5,756  bushels  of  wheat,  1,375  bushels  of  rye, 
8,750  bushels  of  oats,  and  had  room  enough  left  for  2,000  bushels 
more  of  corn.     How  many  bushels  would  the  store-house  hold  1 

A.  23,881  bushels. 

67.  Two  men  started  from  New  York  and  traveled  in  opposite 
directions.  The  one  was  to  go  37  miles  a  day,  and  the  other  35 
miles.     How  far  would  they  be  apart  the  first  night  ?  A.  72  miles. 

68.  How  far  were  they  apart  the  2d  night  1  A.   144  miles. 

69.  How  far  were  they  apart  the  3d  night  1  J..  216  miles. 

70.  How  far  were  they  apart  the  4th  night  1         A.  288  miles. 

71.  How  far  were  they  apart  the  5th  night  ]  A.  360  miles. 

72.  How  far  were  they  apart  the  6th  night  1         A.  432  miles. 


SIMPLE    SUBTRACTION.  39 


SIMPLE  SUBTRACTION/ 

XII.     1.  A  man  owing  9  dollars  paid  3  dollars.  How  many  dol- 
lars did  he  still  owe  ? 

^      (!•)  (2.)  (3.)  (4.) 

From  9  dollars  38  739  9467 

Take  3  dollars  1  5  4  3  2  7  13  7 

*  ^.  6  dollars.  *  *  * 


5.  If  we  take  6  miles  from  27  miles,  we  have  21  miles  left; 
because  21  and  6  are  27:  thus, — 

(6.)  (7.)  (8.)  (9.) 

27  miles.         145         4592         9864        49257 
6  miles.  2  1  3  5  1         2  3  5  1  6  112 

A.  21  miles. 


P.  27  miles. 

10.  Observe  that  like  Addition,  units  must  be  placed  under  units 
tens  under  tens,  hundreds  under  hundreds,  Sfc. 

11.  From  6235  dollars  take  2111  dollars  ?        A.  4124  dollars. 

12.  From  89659  bushels  take  7216  bushels?  A.  82443  bushels. 

13.  A  merchant  having  13,069  bags  of  coffee  sold  9,020  bags ;  how 
many  bags  had  he  remaining  on  hand  ? 

Minuend,"^       13  0  6  9  Say  0  from  9  leaves  9 ;  2  from 

Subtrahend,"^      9  0  2  0  6  leaves  4 ;  0  from  0  leaves  0 ; 

Remainder,        4  0  4  9  9  from  13  leaves  4. 
Proof.             13  0  6  9  A.  4049  bags. 

14.  In  like  manner  perform  and  prove  the  following  examples,— 

(15.)  (16.)  (17.) 

18350000   79527890000   16656213708 
9300000    3121800000    9604000501 


18.  Two  brothers  engaged  in  mercantile  business ;  one  gained 
1,584  dollars  and  the  other  920  dollars.  How  much  did  one  gain 
inore  than  the  other!  A.  628  dollars. 

19.  A  father  gave  36,540  dollars  to  his  son,^^nd  25,000  to  his 
daughter.  How  much  greater  was  his  son's  portion  than  the 
daughter's?  A.  11,540  dollars. 

20.  Subtract  13  million  from  27  million. A.   14,000,000. 

XII.  Q.  How  are  units,  tens,  &c.,  to  be  placed  in  subtraction  ?  10.  Why 
loes  6  from  27  leave  21  ?  5.  How  do  you  subtract  0  from  9  ?  13.  0  from  0  ?  13. 
9  from  13  ?  13.  In  subtracting  9,020  from  13,069  which  number  is  the  minuend  ? 
13.  Which  is  the  subtrahend?  13.  What  is  the  answer  called?  13 

1  Subtraction,  [L.  subtractio.2  The  taking  a  part  from  the  rest. 

2  Minuend,  [L.  minuendus.']  To  be  diminished  or  lessened. 

3  Subtrahend,  from  subtrahendus  L.  to  be  taken  from. 

*  (1.)  A  6  dollars.    (2.)  A.  23.    (3.)  307.    (4.)  A.  2,330. 


40  ARITHMETIC, 

21.  Subtract  85  thousand  from  96  thousand  A.   11,000. 

22.  What  is  the  diflerence  between  one  million  eight  hundred 
thousand,  and  six  hundred  thousand?  A.  1,200,000. 

23.  A  man  bought  a  chaise  for  262  dollars,  and  a  harness  for  39 
dollars.     How  much  did  one  cost  more  than  the  other  1 

We  can't  take  9  from  2,  but  we  can  take  1  ten, 
which  is  10  units,  from  6  tens.  The  10  units 
(borrowed)  added  to  the  2  units  make  12.  Then  9 
from  12  leaves  3.  Now  instead  of  calling  the  6  a 
5,  we  may  add  the  1  ten,  which  we  borrowed,  to  the 
3  tens,  the  next  lower  figure,  since  it  can  make  no  difference  in  the 
result,  for  4  from  6  is  the  same  as  3  from  5.  Therefore  say,  1  to 
carry  to  3  is  4,  which  from  6  leaves  2,  &c. 

24.  Hence,  when  the  lower  figure  is  greater  than  the  one  above 
it,  add  10  to  the  upper  figure^  and  only  10  in  any  case,  then  take  the 
difference,  being  particularly  careful  to  carry  1  to  the  next  lower  figure. 

(25.)  (26.)     (27.)  (28.)  (29.) 

From  356  783  5237  14657  15782 

Take   49  147  1018  3901  190 
A.   3  0  7 


2 

6 
3 

2 
9 

A. 

2 

2 

3 

P. 

2^ 

6  2 

30.  If  we  take  4,508  from  67,297,  how  many  will  remain  1 
(31.)  (32.)  (33.) 

67297         3578634        45632858 
4  5  0  8         2  7  10  9  16        2  9  0  4  19  3  9 

A.  6  2  7  8  9 


33.  A  gentleman  purchased  a  farm  for  4,000  dollars,  and  paid  5 
lollars  as  "  earnest  money." ^     How  much  did  he  still  owe  1 

4  0  0  0  Say  5  from  10  leaves  5 ;  1  to  carry  from  10 

5        leaves  9 ;  1  to  carry  again  from  10  leaves  9  ;  1  to 

A,  3  9  9  5        carry  from  4  leaves  3. 

34.  Since  in  the  last  example,  there  are  no  tens  nor  hundreds  in 
the  upper  line,  we  do  in  reality  borrow  the  10  from  the  4  thousands. 

35.  For  if  we  call  the  4  thousands  10  less,  making  3990,  and  sub- 
tract the  5  without  carrying  any,  the  result  will  be  the  same  as  before. 

36.  Hence  the  10  is  always  borrowed  from  the  first  significant 
figure  on  the  left  in  the  upper  line. 

37.  From  10,000  cents  take  1  cent.  A.  9,999  cents. 

38.  From  10,000  mills  take  9,999  mills.  A.  1  mill. 

Q.  In  taking  39  from  2C2,  how  do  you  subtract  the  9  from  2  ?  23.  Do  you 
borrow  the  10  units  from  the  upper  figure  6,  or  the  lower  figure  3?  23.  To  which 
figure  do  you  pay  it  again,  or  carry  it  ?  23.  How  happens  it  that  you  borrow 
from  one  and  pay  to  another  ?  23.  How  then  do  you  proceed  in  all  such  cases  ? 
24.  In  subtracting  5  from  4000,  for  instance,  from  what  is  the  10  borrowed,  and 
why?  34.  From  which  figure  is  the  10  always  borrowed  {  36. 

1  Earnest.  Seriousness ;  a  reality ;  first  fruits  ;  money  given  to  bind  a  bargain. 


A. 

8  years. 

A. 

4  years. 

A. 

8  years. 

A. 

8  years. 

A. 

8  years. 

A. 

4  years. 

A. 

8  years. 

A. 

4  years. 

SIMPLE    SUBTRACTION.  41 

39.  How  much  does  110,001  exceed  99,999  ]  A.  10002. 

40.  How  much  does  1,234,567  lack  of  2  miUion?     A.  765,433. 

41.  How  much  greater  is  1  million  than  1,0001      A.  999,000. 

42.  How  much  smaller  is  5,000  than  500,000  T       A.  495,000. 

43.  Find  how  much  must  be  added  to  fifteen  thousand  and  five  to 
Aiake  twenty-three  thousand.  A.  7,995. 

44.  The  deluge  happened  A.  M.^  1656,  and  our  Saviour  was  born 
A.  M.  4,004.  How  many  years  intervened  between  these  two 
events  1  A.  2348  years. 

45.  How  many  years  from  the  birth  of  our  Saviour,  to  the  dis- 
covery of  America  by  Columbus,  which  occurred  A.  M.  5,496 1 

A.  A.  D.»  1492. 

46.  Find  by  the  following  list,  the  length  of  time  each  President  of 
the  United  States  held  that  office. 

47.  George  Washington,  from  1789  to  1797. 

48.  John  Adams,  from  1797  to  1801. 

49.  Thomas  Jefferson,  from  1801  to  1809. 

50.  James  Madison,  from  1809  to  1817. 

51.  James  Monroe,  from  1817  to  1825. 

52.  John  Quincy  Adams,  from  1825  to  1829. 
63.  Andrew  Jackson,  from  1829  to  1837. 

54.  Martin  Van  Buren,  from  1837  to  1841. 

55.  William  Henry  Harrison,  from  March  4th,  1841 .      A. 

RECAPITULATION. 

56.  Subtraction  is  the  taking  of  a  less  number  from  a  greater. 

57.  Subtraction  then  is  exactly  the  reverse  of  Addition. 

58.  The  Minuend  is  the  greater  number,  and  the  one  from  which 
the  subtraction  is  to  be  made. 

59.  The  Subtrahend  is  the  smaller  number,  and  the  one  which  is 
to  be  subtracted. 

60.  The  Remainder  is  the  difference  between  any  two  numbers. 

61.  Simple  Subtraction  is  the  subtracting  of  one  number  from 
another,  when  both  are  of  the  same  denomination. 

RULE. 

62.  Place  the  less  number  under  the  greater,  so  that  units  may 
stand  under  units,  tens  under  tens,  c^c. 

63.  Begin  on  the  right  and  take  each  figure  separately,  from  the 
figure  over  it. 

64.  If  the  upper  figure  be  too  small,  add  10  to  it,  then  subtract,  and 
carry  1  to  the  next  lower  figure. 

Q.  What  is  Subtraction?  56.  Does  it  resemble  Addition?  57.  What  is  the 
Minuend?  58.  Subtrahend?  59.  Remainder?  60.  Simple  Subtraction?  61. 
Rule  ?  62,  63,  64.  Order?  65.  Proof?  66.  Reason  of  the  rule?  67. 

1  A.  M.  The  A.  stands  for  Anno,  L.,  meaning,  in  the  year,  and  the  M.  for  Mundi,  of 
the  world ;  hence  A.  M.  1656,  means  so  many  years  since  the  creation  of  the  world. 

2  A.  D.  in  the  year  of  our  Lord,  See  xi.  45. 

4* 


42  ARITHMETIC. 

65.  Order.     Write  down  :  subtract  :  and  carry. 

66.  Proof.     Add  together  the  remainder,  and  subtrahend,  and  if 
their  sum  equal  the  minuend,  the  work  is  right. 

67.  The  rule  and  proof  proceed  both  on  the  same  general  principle, 
viz. ;  "  That  the  sum  of  the  parts  is  equal  to  the  whole^ 

(68.)  (69.)  (70.) 

45678937    42354007    65001039 
11709002     5342109       273420 


71.  From  one  million  take  999.  A.  999,001. 

72.  From  one  million  take  nine.  A.  999,991. 

73.  What  number  added  to  four  hundred  and  fifty-nine  thousand 
will  make  one  million'?  A,  541,000. 

(74.)  (75.) 

5003278009101730  6812345670809080 

4917509080390950  1000900900900990 


76.  Subtract  37,408  from  197,000.  A.  159,592. 

77.  Suppose  a  man  is  worth  100,000  dollars  lacking  1000  doUars. 
How  much  is  he  worth?  ^.99,000. 

78.  Subtract  376,000  from  9,567,000.  A.  9,191,000 

79.  A  father  divided  his  estate,  amounting  to  75,000  dollars, 
between  his  son  and  daughter,  giving  to  the  former  37,560  dollars. 
How  many  dollars  did  he  give  to  the  daughter  1  A.  37,440  dollars. 

80.  A  merchant  bought  goods  worth  125,000  dollars,  on  four 
months  credit,  but  is  offered  a  discount^  of  825  dollars  for  cash  down. 
What  would  be  the  cost  of  the  goods  for  cash  ?   A.   124,175  dollars. 

81.  Suppose  you  buy  goods  worth  35,765  dollars  on  six  months 
credit,  and  are  offered  the  same  goods  for  35,565  dollars  ready 
money.^    What  is  the  discount  offered  you  ?  A.  200  dollars. 


MISCELLANEOUS   EXAMPLES. 

(1.)  ^  (2.) 

Xni.  Add  One  hundred.      Add  Six  hundred  and  twenty-seven. 

Two  thousand,  Two  thousand  and  fifteen. 

Two  million,  Thirty  thousand  and  seventy. 

Four  million,  Forty-two  million  and  eighty, 

Ten  thousand,  Fifty-one  thousand  and  nine, 

Ten  million,  One  billion  one  million  and  one. 

Forty  million,  Seventeen  trillion  and  seventeen 

A.  56,012,100  A.  17,001,043,  oTs  ,819 

1  Discount,  Deduction  ;  allowance  for  prompt  pay. 
S  Ready  Money.  Cash  down  ;  money  paid  down. 


SIMPLE    iJiaTIPLICATION.  43 

(S.) 

chaHtaMe  Objects.     Hr^u^ht^  ^^  " t^  ,tlt'iT   '" 

7.  A  merchant  owing  2,345  dollars  naid  1  40^^"  n'^^^  '^''"''''^- 
ana  M9  ,„,,„.  .,  .„„,V     Ho.  3to';e^°«,1^1'-rr™; 

10.  SnPPOfthedistancefromBostontoNeivYorkbeSin'mn;. 
tlience  to  Philadelphia  90  miles,  and  thence  to  Bakimore^M  m^es 
how  many  miles  are  Philadelphia  and  Baltimore  from  Boston"  ' 

11     rp,  ,    .  ^-  Pa,  300  miles  :  Be.  400  milp^ 

in    «.7^  population  of  the  United  States  in  1820,  was  9  6^8  IfiR  • 

ip^^;:-=ri^or^^^ 

the  increase  of  population  from  the  first  period  of  tTme  to   Lc^^^^^^ 
W  the  second  penod  to  the  third  ;  and  from  the  first  pSote 
•  '^-  3,220,504;  5,141,330;  8,361,834. 


SIMPLE  MULTIPLICATION. 

^q^doLrf  ^^°''  ^  man  earns  9  dollars  a  week,  for  5  successive 
9  dollars;  '  ^""'^  ""^"^  ^""^^'^  ^^^^  ^^  ^^^"  i«  all  ? 

9  dollars.  (1.) 

9  dollars.  Multiplicand.  9  dollars. 

9  dollars.  Multiplier.        5  times. 

A.  45  dollars.  Product.        45  dollars.  A. 

^^p^s\::J2.r.hr'fM°uT.ipTint9\y°^ 

numbers  severally  called  ?  1.  ^"^""iP^ymg  y  by  5  making  45 ;  what  are  the 

^^a^gfven'nuS^S^"^'^^'^^^''-^  ^'^  ^''  «^  multiplying  or  of  increasing  any 


4#  ARITHMETIC. 

2.  Suppose  a  fa,rmer  has  3  pastures  with  502  sheep  in  each  pasture, 
how  many  sheep  will  they  all  make  1 

(2.)  (3.)  (4.)  (5.) 

5  0  2  sheep.     9402     520300  2     601020101 
3  times.  2  3  4 


1 

5  0  6 

sheep. 

8     . 
1  2 

1  2 

8 

9  5 

9  6 

(6.)  When  two  numbers  are  to  be  multiplied  together, 
if  either  be  made  the  multiplier,  the  product  will 
be  the  same ;  but  it  is  more  convenient  to  make 
the  smaller  one  the  multiplicand. 

7.  Recollect  to  write  units  under  units,  tens  under  tens,  4"C. 

8.  Suppose  a  steamboat  receive  702  dollars  for  every  trip  she 
makes ;  when  she  has  made  3  trips,  how  many  dollars  will  she  have 
received  ?  A.  2,106  dollars. 

9.  If  a  man  earns  3  dollars  a  day,  how  much  will  he  earn  in  a 
year,  which  contains  313  working  days  A.  939  dollars. 

10.  If  8,011  men  could  build  a  bridge  in  9  days,  how  long  would 
it  require  1  man  ^lone  to  do  it,  working  at  the  same  rate  1 

A.  72,099  days. 

11.  If  a  man,  traveling  9  hours  a  day,  performs  a  journey  in  210 
days,  how  many  hours  is  he  in  doing  it?  A.   1,890. 

12.  An  agriculturist^  sold  8,101  trees  "of  the  genuine  Chinese 
Morus^  Multicaulis'"^  species,*  for  8  cents  a  tree.  What  sum  did  he 
get  for  the  whole  1  A.  64,808  cents. 

13.  There  are  365  days  in  one  year;  how  many  days  then  are 
there  in  5  years  1 

3  6  5  ^^y  ^  times  5  are  25  ;  set  down  the  5  and  carry 

5         2,  as  in  Addition.     Next  say  5  times  6  are  30  and  2 
A.  18  2  5         to  carry  are  32  ;  write  down  the  2  and  carry  3  as 
■  before,  and  so  on. 

14.  Hence  observe  the  same  rule  for  carrying  as  in  Addition. 

15.  How  many  are  2  times  9,313,654'?  A.  18,627,308. 

16.  How  many  are  3  times  8,252,180 1  A.  24,756,540. 

17.  How  many  are  4  times  3,008,309 1  A.   12,033,236. 

18.  How  many  are  5  times  8,090,876 1  A.  40,454,380. 

Q.  Wliich  number  is  generally  made  the  multiplier  and  why  ?  6.  Give  an  ex  • 
ample,  See  6.  How  are  the  numbers  to  be  written?  7.  In  multiplying  365  by  5 
how  do  you  commence  ?  13.  How  many  do  you  carry  for  25  ?  See  13.  What  is 
done  with  the  2  "to  carry,"  in  nmltiplying  6  by  5?  13.  What  is  the  general 
direction?  14. 

1  Agkicultubist,  flrom  ager,  L.  afield,  and  cultura,  L.  culture.  One  who  tills  the 
ground. 

2  MoRus,  [L.  monis.l  The  mulberry-tree. 

3  MuLTicAULis.  from  multi,  L.  many,  and  caulis,  L.  the  stalk  or  stem.  Henc9 
tnulticauUs  means  having  many  stalks  or  stems. 

4  Species,  L.  Form;  outward  shape ;  sort;  kind. 

*(3.)  A.  18,804.    (4.)  A.  15,609,006.    (5.)  A.  2,404,080,404. 


SIMPLE    MULTIPLICATION.  45 

19.  How  many  are  6  times  7,130,028  ?  A    42  7ftn  iRft 

20.  How  many  are  7  times  6,087  695  ]  J  i/eTsltf 

21.  How  many  are  8  times  4,795,732  ^  2  stsfi's'e. 

RECAPITULATION. 
JtJ^:^^^Z^:Z::^^^''-^  number.  „a„,ti™es 

25 '  Th!  ^^^^^^^^^^^^  i«  the  number  to  be  repeated. 

25.  The  Multiplier  IS  the  number  by  which  we  multiply 

26.  The  Product  IS  the  result  obtained  by  multiplying.'  "^ 

^7.    Ihe  J^  ACTORS'  of  any  number  are  such  numbers  as  will  W 

^^^r^Tl^T^'^t-'^  ''--  theracti^r^^'^^ 

ber'by  SlXr."''"'''''"""^  "  '''  multiplying  of  one  simple  num- 
XV.  When  the  multiplier  does  not  exceed  12. 

1.   Write  the  smaller  number  under  the  greater,  so  that  units  mav 
stand  under  units,  tens  under  tens,  <Sfc  ^ 

by\^Zn-1-  '^''  ^^-°'^lr^  ^"^^'^^y  each  figure  of  the  multiplicand 
by  the  multiplier  separatehj,  carrying  as  in  Addition. 

a'   ?^r^^^'      y^^^"^^    DOWN  :    MULTIPLY  :    AND  CARRY. 

4.   What  IS  the  product  of  40,936  multiplied  by  9  ] 

4  0  9  3  6         5    Multiply  52,031  by  10.  A  520  310 

-———-_         6.  Multiply  67,098  by  11.  A.  738'o78' 

^~±^^^         7.   Multiply  60,359  by  12.  A.  lllfol 

Wh'.tifr'';!'?^"^''  ^?^  ^'^^"'  ^^^'^"^^  ^^'  8  dollars  a  firkin. 

9    T^P..  .      it i^r  ^™  •  ^-  968  dollars. 

ther'e  Inllm^       I     ^'^  "'  "^"'^  '"-^"  '  ^"^^'  "^^"^  ^°"^^«  then  are 
there  in  36,089  eagles  J  .4.  360,890  dollars. 

In  nwu  ^^^f  ^^'^^^  ""^^^"^^  circumference;  how  many  fur- 

longs will  they  make,  reckoning  8  furlongs  to  a  mile  ^     ^i    200  000 
11.  A  ftither  distributed  all  his  property  equally  among  11  sons 

?a  herWafe''  "'"'  ''''''  '°'^"-     '''^^^^  ^^'^^  ^^^  ^-^-  ^^  ^^" 

lauier  S  estate  '  a      nr.O  acc    ^^^^ 


Q.  What 


What  y^T  M  ^1^"\^'Pl^«j'f/«n  ?  22.    What  rule  does  it  abrid.ro,  and  M'hen  ^    23 

be  consKlerod  factorsT  28:  ^?Lus  SLple  SultfplioTtSlS"^  "^^^P^^^^^ 
wSlislS^;ri^?!iyfr  "'  ""''  '''  '°"  '^  y-'P--^d  ?  I,  2. 


4  6 
3 

9 

6 

1 

2 
4 

8  1 
0  7 

4 

1 

6 

8  8 

4 

-4€  ARITHMETIC. 

12.  There  are  12  months  in  a  year.  How  many  months  has  i 
been  since  the  creation  up  to  A.  D.  1840,  making  in  all  A.  M.  5,844 
years]  A.  70,128  months. 

XVI.     When  the  multiplier  exceeds  12. 

1.  Suppose  a  drover  has  bought  469  cows  for  36  dollars  apiece, 
what  must  he  have  paid  for  the  whole  *? 

We  multiply  by  the  6  as  before,  also  by  the  3 ; 
but  observe  that  the  7  is  placed  in  the  tens'  place, 
because  the  3  by  being  3  tens,  is  ten  times  greater 
than  if  it  were  3  units.  Lastly,  add  2,814  and 
1,407  together  as  they  stand,  thus,  4  is  4 ;  7  and  1 
are  8,  &c. ;  for  it  takes  both  the  product  of  6  units 
and3tensor30,  to  make  the  total  product  of  36.  A.  16,884  dollars. 

2.  Observe  that  the  first  figure  in  each  product  is  placed  directly 
under  the  figure  by  which  you  are  multiplying. 

3.  What  is  the  cost  of  675  barrels  of  pork  at  23  dollars  a  barrel  1 

A.   15,525  doDars. 

4.  What  will  37  horses  cost  at  125  dollars  apiece  1     A.  4,625. 

5.  In  one  year  there  are  365  days ;  how  many  days  then  are  there 
in  4,167  years'? 

6.  Multiply  5,426  by  423.  ^.2,295,198. 

7.  Multiply  9,132  by  239.  ^.2,182,548. 

8.  Multiply  6,799  by  425.  J..  2,889,575. 

9.  Multiply  8,706  by  359.  A.  3,125,454. 

10.  Multiply  9,508 by 698.  ^.6,636,584. 

11.  Multiply  8,903  by  452.  jl.  4,024,156. 

12.  Suppose  a  certain  army  of  84,050  men  are  to  receive  their 
year's  pay,  being  153  dollars  apiece  ;  how  much  will  they  all  receive  ? 

A.   12,859,650  dollars. 

13.  If  3,675  men  will  do  a  piece  of  work  in  327  days,  how  many 
men  will  be  required  to  do  it  in  one  day  1  A.  1,201,725  men. 

GENERAL    RULE. 

14.  Multiply  by  each  significant  figure  of  the  multiplier  separately , 
placing  the  first  figure  in  each  product  directly  under  the  figure  by 
which  you  are  multiplying. 

15.  Then  add  together  these  partial  products  in  the  same  order  as 
they  stand ;  the  amount  will  be  the  total  product  required. 

XVI.  Q.  In  multiplying  469  by  36,  how  do  you  proceed  with  the  6  ?  1.  How 
with  the  3?  1.  Why  i.s  the  first  figure  in  the  second  product  placed  in  the  tens' 
place  1  1.  What  is  to  be  done  with  these  partial  products?  1.  What  is  the  direc- 
tion for  placing  each"  product,  when  there  are  several  figures  in  the  multiplier  ? 
2.  What  is  the  General  Rule  ?  14,  15. 

1  Partial,  {"L.  pars.]  Biassed  to  one  party ;  affecting  a  part  only '  not  total. 


4 

1 
3 

6 
6 

7 
5 

2 

0 

8 

3 

5 

2 

5 

0 

0 

2 

1 

2 

5 

0 

1 

.  1 

5 

2^ 

0_ 

9^ 

_5^ 

5 

SIMPLE    MULTIPLICATION.  47 

16.  Multiply  8  5  6  7  8  9  17.  Suppose  3,756,701  to  be  a 

-i~~~^      multiphcand,  and  34,005  a  multi- 
5  14  0  7  3  4  ^       '  "^^^^  "^"^  ^^  ^^^  P^«d»«t^ 


2570367  ,0    .^r.    ^-   127,746,617,505. 

8  5  6  7  8  9  ]^'   When  the  multiplier  is  6,035 

A.   1165661434  5  ^"?,V^^"^"^tiplicand  732,006;  what 

^  ^  wiU  be  the  product  ? 


.    20.  There  are  1,728  solid  inehes  in  I  solidtoff  how  rSny  solid 
yafi'  *■  "="='•  """t^^^ing  25  yards,  how  many 


23.  mat  WiU  be  the  cost  of  7  pieces  of  cloth,  each  pi Jcetn- 
tammg  25  yards  at  12  dollars  per  yard  1  A.  2,100  Mars 

Jl\^^^^^  "''"  "  ''"'*'"  fi'''''  has  475  hills  of  potktoes,  and  that 
wonWafi^'ov,"^^^  "  r^'""^'  how  many  potatoes  at'that  a^e 
would  a  field  275  times  as  large  contain?    A.  1,436,875  potatoes 

24.  Suppose  a  father  can  hoe  in  one  day  3  times  as  many  hills  of 

hrin^fs  dTy"'  "'°  '"'  ''"  •""'  • '"'"  ""^  ""'^  coSefol"' 

25.  If  25  men  dig  a  ditch  in  375  days,  how  4  wm'ftake'one 
man  alone  to  dig  45  such  ditches  ?  ^    421  875  ^7 

26.  There  are  6  days  in  the  week  for  labor  and  study ;  suppose 

how  many  kernels  -^i  the  baskets  contain  in  aU  ?     A.  1,256,000     ' 

^^¥'f7^  "^"^*^P^^  ^y  ^  composite^  number. 
1.  A  Composite  Numbers  one  that  is  produced  bv  multivhn^rr 
any  two  smaller  numbers  together.  ^  multiplying 

the  6  and  7  its  factors,  or  as  they  are  sometimes  caUed,  its  component 

hp^  J"  '^^  T"^^^^'  ^  *^''  ^  ^''^  ^'  ^^^  5  i«  ^«t  ^  composite  num- 
ber because  there  are  no  two  smaller  numbers  that,  multiplied  to- 
gether,  will  produce  it.  7  i-         v 

4.  A  Composite  Number  may  have  two,  three,  or  more  factors; 
for  3  times  4  are  12,  and  8  times  12  are  96.  Here  96  is  composed 
ol  the  factors  3,  4  and  8. 

1  fV\  ^P^^f  '^  ^  composite  number?  1.    Give  an  example.  2.    Why  are 


43  ARITHMETIC. 

5.  The  finding  what  two  or  more  numbers  multiplied  together  will 
make  a  given  number,  is  called  resolving^  it  into  factors. 

6.  What  will  24  carriages  cost  at  375  dollars  apiece. 
3  7  5 

Z  The  first  product  is  the  cost  of  4  carriages  ;  then  6 

15  0  0  times  that  product  must  be  the  cost  of  24  carriages, 

6  for  4  times  6  are  24.                         A.  9000  dollars. 

9  0  0  0 


7.  In  the  last  example,  any  other  factors  would  answer  the  same 
purpose ;  as,  2  and  12,  or  3  and  8,  or  2,  3,  and  4. 

Note.  When  composite  numbers  do  not  exceed  144,  their  factors 
are  easily  found  by  the  Multiplication  Table. 

RULE. 

8.  Resolve  the  comfosite  number  into  two  or  more  factors,  and 
multiply  by  them  successively.'* 

9.  Multiply  670,032  by  18.  A.  12,060,576. 

10.  Multiply  724,081  by  27.  A.  19,550,187. 

11.  Multiply  634,081  by  35.  A.  22,192,835. 

12.  Multiply  754,038  by  96.  A.  72,387,648. 

13.  Multiply  603,407  by  108.  A.  65,167,956. 

14.  Multiply  708,936  by  144.  A.  102,086,784. 

15.  Multiply  608  by  24,  using  three  factors  A.   14,592. 

16.  Multiply  37  by  120,  using  four  factors.  A.  4,440. 

XVIII.  To  multiply  by  10,  100,  1000,  «&c. 

RULE. 

1.  Annex  to  the  multiplicand  all  the  ciphers  in  the  multiplier.  See 
X.   15. 

2.  Multiply  68,345  by  10.  Annex  1  cipher.  A.  68,3450. 

3.  Multiply  3,45    by  100.  Annex  2  ciphers.  A.  34,500. 

4.  Multiply  678  by  1000.  Annex  3  ciphers.  A.  678,000. 

6.  Suppose  53,467  to  be  a  multiplicand,  and  10,000  a  multiplier, 

what  will  be  the  product  1  A.  534,670,000. 

6.  Multiply  63,456  by  10.  A. 

7.  Multiply  38.065  by  100.  A. 

8.  Multiply  65,721  by  1000.  A. 

9.  Multiply  37,561  by  10,000.  A. 

10.  Multiply  91,509  by  100,000.  A. 

11.  Find  the  sum  of  the  last  5  answers.        A.  9,596,672,060. 

12.  There  are  10  mills  in  1  cent,  10  cents  in  1  dime,  10  dimes  in  1 
dollar,  and  10  dollars  in  1  eagle  ;  how  many  mills  then  are  there  in 
678,345  eagles'? A.  6,783,450,000  miUs. 

Q.  What  is  tlie  finding  of  factors  called  ?  5.  How  do  you  multiply  by  the 
factors  of  2'i  ?  6.  What  is  the  Rule  ?    8.  ,       „„     ,« 

XVII  [.  Q.  How  can  you  multiply  by  10,  or  100,  or  1000  easily  1  Why  ?  See 
X.  15. ____» 

1  Resolving.  Separating  into  component  parts  j  analyzing ;  determining. 

2  Successively,  One  after  another  regularly. 


SIMPLE    MULTIPLICATION  49 

XIX.  When  there  are  ciphers  on  the  right  of  either  or  both  of  the 
factors. 

1.  All  such  terms  are  resolvable^  into  two  factors,  one  of  which 
will  always  be  either  10,  or  100,  or  1000,  &c. 

2.  Thus,  the  factors  of  890  are  89  and  10  ;  of  3500,  they  are  35 
and  100. 

3.  Instead,  however,  of  proceeding  as  in  xvii.,  we  can  abbreviate* 
the  process  by  xviii.  as  follows. 

RULE. 

4.  Multiply  only  the  significant  figures  together,  and  annex  to  the 
product  all  the  ciphers  on  the  right  of  both  factors. 

(5.) 
3  6  3  0  6.  There  are  100  years  in  a  century,  how 

^  ^  ^  ^  0  many  years  then  in  32  centuries  1 
10  8  9  A.  3,200  years. 

"^  ^  ^ 7.  There  are  3C5  days  in  1  year,  how  many 

8^3  4  9  0  0  0  0  days  then  in  3,200  years?      A.  1,168,000. 

8.  There  are  24  hours  in  1  day ;  how  many  hours  then  in  2,168,000 
daysl  ^.  52,032,000  hours. 

9.  There  are  60  minutes  in  1  hour ;  how  many  minutes  then  in 
52,032,000  hours  1  A.  3,121,920,000  minutes. 

10.  There  are  60  seconds  in  1  minute  ;  how  many  seconds  then  in 
3,121,920,000  minutes  1  A.  187,315,200,000  seconds. 

11.  Suppose  a  canon  ball  flies  1  mil^  in  8  seconds,  how  long  would 
it  be,  at  that  rate,  in  flying  round  the  earth,  it  being  25,000  miles  ? 

A.  200,000  seconds. 


MISCELLANEOUS    EXAMPLES. 

XX.   1.  What  is  the  whole  number  of  inhabitants  in  the  world, 
there  being,  according  to  Hassel,  in  each  grand  division  as  follows ; 
Europe,  one  hundred  and  eighty  millions ; 
Asia,  three  hundred  and  eighty  millions  ; 
Africa,  ninety-nine  millions ; 
America,  twenly-one  millions ; 
Australasia,  &c.  two  millions  1  A.  682,000,000. 

2.  If  one  man  in  a  factory  earns  375  dollars  a  year,  how  many 
dollars  will  345  men  earn  at  that  rate  in  the  same  time?  A.  129,375. 

3.  A  father  deceased,  left  an  estate  of  40,000  dollars  to  his  3  sons 
and  his  widow.  He  directed,  9,750  dollars  to  be  paid  to  each  son. 
What  was  the  widow's  part?  A.   10,750  dollars. 

XIX.  Q.  When  any  number  has  one  or  more  ciphers  on  the  right ;  what  may 
always  be  one  of  its  factors  ?  1.  What  are  the  factors  of  890  ?  2.  Of  3500 1  2. 
What  is  the  direction  for  multiplying  by  such  numbers  ?  4. 

1  Resolvable.  That  may  be  resolved  or  reduced  to  first  principles. 

2  Abreviate.  To  shorten ;  to  make  shorter  by  contracting  the  parts ;  to  reduce  to  a 
smaller  compass. 

5 


50  ARITHMETIC. 

4.  A  gentleman  paid  for  the  building  of  his  house  by  contract, 
7,650  dollars  ;  for  the  site  4,500  dollars ;  he  was  worth  50,000  dollars 
when  he  began  to  build  ;  how  much  had  he  left  ?  A.  37,850  dollars. 

5.  A  farmer  bought  769  sheep  for  3  dollars  apiece.  After  keeping 
them  3  years  they  doubled  their  number.  What  was  their  value  at 
the  price  he  paid  for  the  original  stock  1  A.  4,614  dollars. 

0.  A  merchant  bought  a  bale  of  goods  consisting  of  40  pieces,  and 
each  piece  of  23  yards,  at  7  dollars  a  yard.  What  did  the  bale  cost 
him?  ,  A.  6,440  dollars. 

7.  A  bookseller  shipped  for  New  Orleans  4  boxes  of  books,  each 
box  containing  400  books.  Now  suppose  each  book  had  200  pages, 
and  each  page  45  lines,  and  each  line  43  letters  ;  how  many  letters 
then  were  there  in  the  4  boxes  1  A.  619,200,000  letters. 

8.  A  speculator  took  out  to  the  west  20,000  dollars  with  which  he 
purchased  1,250  acres  of  land  for  2  dollars  an  acre  ;  1,400  acres  for 
3  dollars  an  acre,  and  3,000  acres  for  4  dollars  an  acre.  How  much 
money  had  he  left  on  his  return  ]  A.  1,300  dollars. 


SIMPLE  DIVISION/ 

XXI.   1.  A  man  having  28  dollars,  laid  it  all  out  in  broadcloth,  at 

2  8  4  dollars  a  yard  ;  how  many  yards  could  he  buy  1 

4  i^           2.  For  every  4  dollars  he  bought  1  yard ;  then  he 

2  4  could  buy  as  many  yards  as  there  are  fours  in  28. 

4  IL           3.  Subtracting  on  the  left  4  from  28  leaves  24  :  4  from 

2  0  the  remainder  24,  leaves  20  :  and  continuing  to  do  so 

4  m.      we  find  at  last  that  nothing  remains. 

16  4.  It  appears  by  counting  that  4  is  subtracted  7  times ; 

_4  IV.      then  there  are  7  fours  in  28,  that  is,  4  is  contained  in 

1  2  28,  just  7  times  :  consequently  he  could  buy  7  yards ; 

4_  li      which  is  the  answer. 

8  5.  The  process  however  by  subtraction  would  be,  in 

ji.  yi-      most  cases  exceedingly  tedious,  compared  with  that  by 

4  division. 

4    —  (7.)  (8.) 

=  (6.)  Say  4  in  28, 7  times ;         Proof. 

By  Division.  because  4  times  7  „  Quotient 

Divisor^  4)28  Dividend^         are  28  :    which  4  Divisor. ' 

7  Quotient^.           '"l^Jy^Z'         ^  ^'^i^'^^' 
___^ • 

1  Division,  [L.  divisio.'i  The  act  of  dividing  or  separating  into  parts  ;  a  part  or  dis- 
tinct portion  ;  a  part  of  an  army  or  fleet ;  disunion  ;  discord. 

2  Divisor,  [  L.  divisor.']  A  dividing  number,  or  that  number  which  shows  how  many 
parts  a  given  number  is  to  be  divided  into. 

3  Dividend,  W..  dividendus.]  To  be  divided;  a  share  of  profits  in  banks,  &.c. 

4  Quotient,  [L.  quoties.'^  The  number  showing  how  many  parts  any  thing  is  divided 
Into. 


SIMPLE    DIVISION.  61 

9.  In  like  manner  perform  and  prove  the  following  examples. 
(10.)  (11.)  (12.)  (13.)  (14.) 

6)4  2       8)5  6       9)108       10)110       11)132 

A.l 


15.  Divide  369  dollars  equally  among  3  men. 

Say  3  in  3, 1  time  ;  3  in  6, 2  times ;  3  in  9, 3  times, 

^  )  ^  ^  ^         writing  each  under  that  figure  that  contains  it :  for 

^•123         the  369  really  means  300 ;  60 ;  and  9  :  then  3  in  300 ; 

100  times ;  3  in  00;  20  times;  3  in  9,  simply  3  times. 

16.  Hence,  to  preserve  the  different  values  of  figures  in  the 
quotient,  each  figure  must  he  written  under  the  figure  that  contains  it. 

(17.)         (18.)         (19.)         (20.) 
2)1806   4)36804   7)4907   8)72808 
^.903   . 

21.  A  farmer  sold  a  quantity  of  wheat  for  27,030  dollars,  receiving 
at  the  rate  of  3  dollars  per  bushel.     How  much  wheat  did  he  sell  ] 

A.  9010  bushels. 

22.  How  long  would  it  take  3  men,  working  at  the  same  rate,  to 
do  what  1  man  is  2,709  days  about  ]  A.  903  days. 

23.  How  many  times  is  6  contained  in  480,609  ■?    Thus, — 
6)480609  The  3  over  is  properly  called  the  Remain- 
A.        8  0  1  0  1  f         der,' under  which  is  written  the  Divisor  6. 

6  See  VI.    2.   Proof.     Say  6  times  1  are  6, 

P.    4  8  0  6  0  9  &c.,  and  3  remaining  are  9 ;  6  times  0 

-         are  0,  &c. 

24.  Recollect  to  add  in  the  remainder  m  proving  the  operation. 

(25.)  (26.)  (27.) 

3)27037    4)16009    5)30050054 

28.  How  many  piles  of  4  dollars  in  each  pile,  will  21  dollars  make  ? 

A.  5  piles  and  1  dollar  over  or  remaming. 

29.  If  the  dividend  is  dollars,  the  remainder  will  of  course  be  dol- 
lars ;  if  sheep,  the  remainder  will  be  sheep,  it  being  always  like  the 
dividend. 

30.  Suppose  a  divisor  to  be  3,  the  dividend  to  be  654,  what  will  be 
the  quotient  ? 

XXI.  Q.  What  are  the  two  methods,  for  finding  how  many  times  one  number 
is  contained  in  another?  1,  6.  Which  is  the  shorter  method  ?  See  5  and  G.  How 
many  subtractions  are  necessary  to  find  how  many  times  4  is  contained  in  28  ? 
See  4.  How  is  the  same  result  obtained  by  division?  6,  7.  What  are  the  4 ;  28  and 
7  severally  called  ?  6.  Which  terms  are  multiplied  together  in  the  proof?  8. 
Which  term  in  division  is  reproduced  l)y  the  proof  ?  See  6  and  8.  How  is  each 
figure  of  the  quotient  to  be  written  in  dividing,  and  why  ?  16.  What  denomination 
is  the  Remainder?  29.  What  is  to  be  done  with  it  in  proving  ?  23.  Whcnindivid- 
ing  there  are  left  2  tens,  for  instance,  and  the  next  figure  is  4,  how  do  you  pro- 
ceed?  30.  Why?  30. 

1  Remainder,  [L.  remaneo.]  That  which  is  left  after  the  separation  into  equal 
parts;  relics;  remains;  the  sum  left  after  subtraction  or  any  deduction. 


Hi 


ARITHMETIC. 


3)654 

A. 

2  1  8 
3 

P. 

6  5  4 

Say  3  in  6,  2  times ;  3  in  5,  1  time,  and  2  tens 
over,  which  are  joined  to  the  4,  thus,  24 ;  then  3  in 
24,  8  times ;  for  the  2  tens  being  20  units  make 
with  the  4  units  24  units. 

31.  Recollect  then  not  to  add  lohat  is  over  to  the  next  figure,  but 
TO  PREFIX^  it. 

32.  How  many  times  4  in  96,877  ?  A.  24,219|. 

33.  How  many  times  6  in  39,247?  A.  6,541|. 

34.  How  many  times  7  in  79,189 1  A.  ll,312f. 

35.  How  many  times  4  in  60,300 1 

4)60300  Say  4  in  6,  1  time  and  2  over ;  4. in  20,  5  times ; 

A.    15  0  7  5         4  in  3,  0  times  and  3  over ;  4  in  30,  7  times  and 
■     ~         2  over ;  4  in  20,  5  times. 

36.  How  many  times  8  in  90,246  ?  A.  ll,280f. 

37.  How  many  times  9  in  85,056  ?  A.  9,450|. 

38.  How  many  times  8  in  33,001  ?  A.  4,125|. 

(39.)  (40.)  (41.) 

10)49505    1  1)89047    12)39421 

42.  When  1  bushel  of  wheat  will  buy  12  bushels  of  potatoes; 
how  many  bushels  of  wheat  will  3,705,602  bushels  of  potatoes  buyl 

A.  308,800^2  bushels. 

RECAPITULATION. 

43.  Division  is  finding  how  ifiany  times  one  number  is  contained 
in  another. 

44.  Division  is  exactly  the  reverse"  of  multiplication,  and  a  con- 
cise method  of  performing  many  subtractions. 

45.  The  Dividend  is  the  number  to  be  divided. 

46.  The  Divisor  is  the  number  to  divide  by. 

47.  The  Quotient  shows  the  number  of  times  the  divisor  is  con- 
tained in  the  dividend. 

48.  The  Re]mainder  is  always  less  than  the  divisor,  and  of  the 
same  denomination  with  the  dividend. 

49.  Division  may  be  performed  by  subtracting  the  divisor  from  the 
dividend  as  many  times  as  there  are  units  in  the  quotient. 

50.  Since  the  ^/u'wor  and  quotient  multiplied  together  will  repro- 
duce^ the  dividend,  therefore, — 

51.  The  divisor  and  quotient  correspond  to  the  factors  of  Multi- 
plication,  and  the  dividend  to  the  product. 

Q.  What  direction  is  given  in  such  cases  ?  31.  What  is  Division?  43.  What 
is  said  of  Division  in  respect  to  Multiplication  and  Subtraction  ?  44.  What  is 
the  Dividend?  45.  Divisor?  46.  Quotient?  47,  Remainder?  48,  How  may 
Division  be  performed?  49. 

1  Prefix,  [L.  prefigo."]  To  put  or  fix  before  or  at  the  beginning. 

2  Reverse,  [L.  reversus.'^  Change;  vicissitude;  a  turn  of  afltiirs ;  misfortune;  a 
contrary;  opposite. 

3  RErRODUcE,  [re  and  prodwco.]  To  produce  again  ;  to  renew. 


6WFLE    DIVISION.  53 

52.  Hence  the  Proof.  Multiply  the  divisor  and  quotient  together, 
and  add  in  the  remainder  ;  if  this  result  be  equal  to  the  dividend  the 
Vvork  is  right. 

53.  Simple  Division  is  the  dividing  of  one  simple  number  by  an- 
other. 

XXII.  Short  Division  is  when  the  Divisor  does  not  exceed  12. 

RULE. 

1.  Having  icritten  the  divisor  on  the  left^  find  hoio  many  times  it  is 
contained  in  the  first  figure,  or  figures  of  the  dividend,  and  write  it 
underneath,  in  the  place  of  the  quotient. 

2.  When  any  figure  is  too  small,  or  when  there  is  a  remainder, 
prefix  it  to  the  next  figure,  then  divide  as  before,  supplying  the  vacant 
places  of  the  quotient  ivith  ciphers. 

3.  Under  the  last  remainder  lorite  the  devisor,  with  a  line  between, 
and  annex  it  to  the  quotient. 

(4.)  (5.)  (6.) 

2  )  3  7  8  5  C  7  0         3)3785670         4)8037019 
.4.   1  8  9  2  8  3  5 


7.  A  man  purchased  sheep  at  4  dollars  a  head,  and  laid  out  1,020 
dollars.     How  many  sheep  did  he  buy  t  A.  255  sheep. 

8.  An  Englishman,  returning  from  a  tour,  found  his  whole  ex- 
penses 1000  guineas,  at  an  average  expense  of  3  guineas  a  day; 
how  many  days  must  he  have  spent  in  traveling!      A.  333^  days. 

9.  Suppose  a  railroad  car  goes  7  miles  while  a  footman  is  going  1 
mile ;  how  far  will  the  footman  go  while  the  car  is  going  37,856 
miles  1  A.  5,408  miles. 

10.  A  father  gave  to  his  youngest  son  8,506  dollars,  which  was  8 
times  the  value  of  what  the  oldest  had.  What  was  the  portion  of 
the  oldest  son  ■?  A.   1,063|  dollars. 

11.  Suppose  an  apprentice  is  9  times  as  long  in  performing  the 
same  piece  of  work  as  a  journeyman ;  how  long  will  it  take  the  latter 
to  accomplish  what  the  former  does  in  1,460  daysl  A.   162f  days. 

12.  Suppose  a  man  gives  away  1  dollar  as  often  as  he  makes  10 
dollars ;  when  he  has  accumulated^  608,350  dollars,  how  much  will 
he  have  given  away  1  A.  60,835  dollars. 

13.  Suppose  the  capital  of  a  bank  amounts  to  5,000,000,  dollars 
and  that  it  is  equally  owned  by  1 1  men  ;  what  is  each  one's  interest 
in  the  bank  1  A.  454,545yV  dollars. 

Q.  With  what  other  ternls  of  a  different  rule  do  the  divisor  and  quotient  Cor 
respond?  51.  With  what  does  the  dividend  correspond?  51.  How  do  we  ascer- 
tain this?  52.  How  many  times  11  in  140?  Which  number  is  the  dividend? 
Which  the  divisor  ?  What  is  the  quotient  ?  What,  the  remainder  ?  What  is 
Simple  Division?  53. 

XXn.  Q.  What  is  Short  Division?  W^hat  is  the  Rule?  1,2.  How  is  the 
remainder  to  be  expressed?  3.   What  is  the  Proof! 

1  Accumulated,  [L.  accumulo.]  Collected  into  a  heap  or  great  quantity. 
5* 


54  ARITHMETIC. 

14.  The  circumference^  of  the  earth  is  about  25,000  miles  ;  what 
is  I  of  that  distance  ?  A.   12,500  miles. 

15.  What  is  -^-  of  25,000  miles  t  A.  8,333^  miles. 

16.  What  is  \  of  25,000  bushels  t  A.  0,250  bushels. 

17.  What  is  |  of  25,000  dollars  1  A.  5,000  dollars. 

18.  What  is  \  of  25,000  ounces  t  A.  4,166^-  ounces. 

19.  What  is  \  of  25,000  pounds!  A.  3,57X3  pounds. 

20.  What  is  |  of  25,000  hours?  A.  3,125  hours. 

21.  What  is  \  of  25,000  days  1  A.  2,1111  days! 

22.  What  is  yV  of  25,000  drams  t  A.  2,500  drams. 

23.  What  is  yV  of  25,000  inches  \  A.  2,272y'V  inches. 
84.  How  much  is  ^^  i    Divide  500  by  7.  A.  71f. 

25.  Howmuchis ''i'^1  A.  52|. 

26.  How  much  is  ^^"^  t  ^    9q7 

27.  What  is  the  value  of  ^-^^  1  A.  4,063y3- 

28.  What  is  the  value  of  '^-^jpi.  A.  3,545||. 

29.  The  salary  of  the  President  of  the  United  States  is  25,000 
dollars  a  year ;  what  is  that  a  month,  allowing  12  months  to  the 
year?  A.  2,083y\  doUars. 

30.  The  President's  salary  then  is  about  2,083  dollars  a  month ; 
what  is  it  a  week,  allowing  4  weeks  to  a  month  I  A.  520f  dollars. 

31.  Allow  the  President's  salary  to  be  just  520  dollars  a  week, 
what  is  it  a  day^  there  being  7  days  in  a  week  t       A.  74^-  dollars. 

32.  Division  has  hitherto  been  performed  partly  in  the  mind,  and 
partly  on  the  slate,  and  is  called  Short  Division,  to  distinguish  it  from 
Long  Division,  in  which  the  divisor  is  so  large  that  we  are  obliged  to 
Write  down  the  whole  operation. 

XXni.  Long  Division  is  when  the  divisor  exceeds  12. 

1.  Recollect  you  are  to  proceed  as  in  Short  Division,  excepting  the 
entire  work  is  to  he  written  out. 

2.  Divide  15  dollars  equally  among  13  poor  persons. 

t\-  ■         1   o  \  r*K  f^  n  ^          Proceeding  as  before,  say  13  in  15, 

Divisor,  13)15(1  Quo.     ,  ^.           i  ^                 ^    ^i.           • 

,             13  ^  ^^"^^  ^^^  ^  over,  write  the  quotient 

*,        .    ,    —r~-  figure  1,  on  the  right  of  15  to  make 

Remainder, J^  ^^^^  ^^^  13  ^^^^^  ^^^  15 .  ^^^^  ^3 

Proof.     13  times  1  are  13,     ^^^  ^^  ^^^^^^  2,  as  at  first 
and  2  are  15,  the  dividend.  ^-   ^^J  dollars. 

3.  Perform  and  prove  the  following  examples  in  like  manner. 
(4.)  (50  (6.)  (7.)  (8.) 

13)16(     14)19(     15)22(     16)25(     17)33( 


XXIII.  Q.  What  is  Long  Division  ?  How  does  it  differ  from  Sliort  Division? 
XXII.  32.  What  is  the  first  direction?  1.  Where  is  the  quotient  written ?  2. 
What  is  the  proof  that  13  is  contained  in  15,  1  time  and  2  over  ? 

1  CiBCUMFERENCE,  [L.  circumferentia.]  The  line  that  bounds  ^  circle;  diitanoo 
round ;  orU  t ;  circle ;  any  thing  circular  or  orbicular. 


SIMPLE    DIVISION.  55 

9.  Recollect  to  place  the  divisor  on  the  left;  the  quotient  on  the 
right;  and  having  multiplied  the  divisor  and  quotient  together^  to 
place  their  product  under  the  dividend,  then  subtract  one  from  the 
other. 

10.  How  many  times  27  in  40 1  A.  U?. 

11.  How  many  times  50  in  811  A.  lf|. 

12.  How  many  times  75  in  96  \  A.   \^. 

13.  At  39  dollars  a  month,  how  many  months*  labor  can  be  pro- 
cured for  75  dollars  1  A.   Iff  months 

14.  What  is  the  quotient  of  39  divided  by  18 '? 
18)39(2  Here  18  is  contained  in  39  ;  2  times,  for  3  times 

3  6  18  are  more  than  39,  and  only  1  time  18  would 

3  leave  a  remainder  larger  than  the  divisor.     See 

==  V.  14.  A.  2y\  acres. 

15.  Divide  41  by  19.  A.  2  and  f\  remainder, 

16.  Divide  70  by  16.  A.  4  and  -/^  remainder. 

17.  Divide  91  by  14.  A.  6  and  -^^  remainder. 

18.  At  13  dollars  a  barrel,  how  many  barrels  of  flour  can  be 
bought  for  99  dollars  ?  A.  Ij^  barrels. 

19.  At  25  dollars  an  acre,  how  much  land  would  103  dollars  buy  1 
a  5  )  1  0  3  (  4  Here  25  is  in  103,  4  times,  for  5  times  25 

10  0  are  more  than  103;  and  only  3  times  25  would 

3  leave  a  remainder  larger  than  the  divisor. 

'  '  A.  4^j  acres. 

20.  Hence  if  the  divisor  and  the  figure  you  put  in  the  quotient  make 
a  product  greater  than  the  dividend  ;  ruh  out  the  work  and  place  a 
smaller  figure  in  the  quotient. 

21.  But  if  the  remainder  is  as  large,  or  larger  than  the  divisor; 
rub  out  the  work  and  place  a  larger  figure  in  the  quotient. 

22.  Divide  108  by  25.  Remainder  -^^. 

23.  Divide  125  by  30.  Remainder  ^. 

24.  Divide  165  by  45.  Remainder  f4. 

25.  Divide  250  by  67.  Remainder  ^. 

26.  Divide  510  by  95.  Remainder  ^. 

27.  Divide  908  by  99.  Remainder  |^. 

28.  Suppose  1,036  pounds  of  beef  are  to  be  shared  equally  among 
125  soldiers,  what  is  each  man's  portion  T 

125)1036(8  '^^  must  alv/ays  take  figures  enough  at 

10  0  0  fii'st  to  contain  the  divisor  once  at  least, 

' 3~g  even  if  it  require  a  hundred, 

■  A. 

29.  Divide  1,039  by  126.  Remainder     31. 

30.  Divide  1,208  by  135.  Remainder  128. 

31.  Divide  2,085  by  250.  Remainder     85. 

32.  Divide  3,780  by  395. Remainder  225. 

Q.  How  can  you  find  what  figure  to  place  in  the  quotient?  20, 21. 


S6  AttlTHMETiCi 

33.  Divide  6,901  by  94L  Remainder  314. 

34.  How  many  building  lots,  at  950  dollars  apiece,  may  be  bought 
for  2,895  dollars  I  A.  SgVir  lots. 

35.  Suppose  a  regiment  contains  1624  men,  how  many  companies 
of  203  men  each  would  be  required  to  make  such  a  regiment  ? 

A.  8  companies. 

36.  Divide  10,835  by    2,083.  Remainder        420. 

37.  Divide  26,008  by     6,041.  Remainder        803. 

38.  Divide  59,346  by    7,085.  Remainder     2,566. 

39.  Divide  50,738  by  20,301.  Remainder  16,136. 

40.  Divide  98,304  by  10,605.  Remainder     2,859. 

41.  Divide  90,090  by  20,303.  Remainder     8,878. 

42.  A  father  divided  an  estate  of  813,824  dollars  equally  among 
his  sons,  giving  to  each  203,456  dollars.     How  many  sons  had  he  1 

A.  4  sons. 

43.  When  flour  is  13  dollars  a  barrel,  how  many  barrels  will  329 
dollars  purchase  ? 

13)329(25  Say  2  times  13  are  26,  writing  the  26  under 

2  Q  the  32.     The  6  over  is  by  Short  Division  to 

Q~Q  be  prefixed  to  9  ;  or  which  is  the  same  thing, 

Q  5  bring  down  the  9,  and  annex^  it  to  the  6 ; 

• — ^  then  13  in  69 ;  5  times,  for  5  times  13  are  08, 

-.  .V  -  and  4  remainder.  A.  25y\  barrels. 

44.  Therefore,  take  at  first  only  figures  enough  to  contain  the 
divisor,  and  having  divided  them,  annex  to  the  remainder  the  next 
figure  of  the  dividend ;  after  which  divide  as  before,  and  so  on  till 
the  figures  of  the  dividend  are  all  brought  down. 

45.  Divide  278  by  13.  A.  Quotient  21.      Remainder    5. 

46.  Divide  985  by  25.  A.  Quotient  39.      Remainder  10. 

47.  Divide  988  by  46.  A.  Quotient  21.      Remainder  22. 

48.  What  is  the  quotient  of  3,369  divided  by  25  T 
26)3369(134  After  having  brought  down  the  6  and 

2  5  divided,  bring  down  the  9  and  annex  it 

8  6  to  the  1 1 ;  which  divide  as  before. 
7  5  A.  134i§. 

119  49.  Divide 3,386 by 25.  ^.Rem.  11. 

10  0  50.  Divide  6,798  by  45.  ^.  Rem.    3. 

1  9  51.  Divide  8,241  by  35.  A.  Rem.  16. 

52.  Wlien  hay  is  25  dollars  a  load,  how  much  can  be  bought  for 
52,186  dollars  1 

Q.  How  many  left-hand  figures  do  you  take  first  ?    44.    When  there  is  a 

remainder  in  dividing,  how  do  you  proceed?  44. 

1  Annex,  [L.  annecto.^  To  unite  at  the  end;  to  subjoin ;  to  affix,  to  connect  with 


2 
2 

1  8 
0  0 

1  8  6 
1  7  5 

1 

2 

SIMPLE    DIVISION.  57 

Since  25  is  not  contained  in  21« 
25)52186(2087        put  a  cipher  in  the  quotient  and 
^  "  bring  down  the  8,  then  divide  as 

before.  A.  2,087^|  loads. 

53.  When  then  the  remainder 
with  one  figure  annexed  is  too 
small ;  mark  its  place  in  the  quO' 
tient  by  a  cipher  and  bring  doicn  an- 
other figure,  then  divide  as  before. 

54.  Divide  209,491  by  102.  Quotient  2,053^. 

55.  Divide  280,483  by  112.  Quotient  2,504yV^. 

56.  Divide  401,123  by  125.  Quotient  3,208|||. 

57.  If  the  remainder,  with  two  figures  annexed,  is  still  too  small , 
annex  another  figure  of  the  dividend  as  before,  and  thus  continue  an- 
nexing till  you  obtain  a  number  large  enough  to  contain  the  divisor. 

(58.)  (59.) 

165)495825(3005  A.  209)6270836(30004  A 
49  5  627 

825  836 

825  836 


60.  Divide  115,611,740  by  2,312.  A.  5,0005J^. 

61.  Divide  345,235,530  by  4,315.  A,  8,0008^f||. 

62.  A  gentleman  expended  1,253,763  dollars  for  land,  paying  for 
each  acre  125  dollars ;  how  many  acres  did  he  buy  T 

Dividend. 
Divisor,  1  2  5)12  53  763(1  0  0  3  OyV^  acres  ;  Quotient. 

63.  What  is  the  quotient  of  95,658  divided  by  245 1  A.  390|^|. 

64.  When  the  divisor  is  103  and  the  dividend  42,024,  what  is  the 
quotient?  A.  408. 

65.  When  the  quotient  is  408  and  the  dividend  42,024,  what  is  the 
divisor?  A.  103. 

616.  When  the  divisor  is  103  and  the  quotient  408,  what  is  the 
dividend?  (103  times  408).  A.  42,024. 

67.  When  the  quotient  is  408,  the  divisor  103,  and  the  remainder 
98,  what  is  the  dividend  ?  A.  42,122. 

68.  When  the  dividend  is  42,122,  and  the  divisor  103,  what  is  the 
quotient  and  remainder  ?  A.  408fVV 

69.  The  salary  of  the  President  of  the  United  States  is  25,000 
dollars  per  annum ;  what  is  that  a  day,  allowing  365  days  to  the  year] 

A.  68if§  dollars. 

GENERAL  RULE. 

70.  Begin  on  the  left-hand  of  the  dividend.,  and  take  the  fewest 
figures  that  will  contain  the  divisor,  and  write  the  number  of  times  it 
is  contained  in  them  on  the  right  of  the  dividend,  for  the  first  quotient 
Hgure. 

71.  Multiply  the  divisor  by  this  quotient  figure,  and  place  their  pro- 


fi8  ARITHMETIC. 

dttct  under  the  figures  of  the  dividend  used,  subtract  it  therefrom  and 
annex  to  the  remainder  the  next  figure  of  the  dividend,  which  divide 
as  before,  and  so  on. 

72.  But  if  the  remainder  thus  increased  is  still  too  small,  write  a 
cipher  in  the  quotient  and  annex  another  figure,  and  so  on  till  it  does 
become  large  enough. 

73 .  When  the  product  of  the  divisor  and  quotient  figure  is  too  large, 
the  latter  must  be  diminished ;  but  when  the  remainder  is  as  large 
or  larger  than  the  divisor,  the  quotient  figure  must  be  increased. 

74.  Order.     Find  how  many  times  ;  multiply  ;  subtract;  and 

BRING  down. 

75.  Proof.     The  same  as  in  Short  Division. 

76.  Divide  14,150  by  115.  Quotient     \2^j\j. 
11.  Divide  28,682  by  121.  Remainder      y|y. 

78.  Divide  2,360,557  by  1,021.  Remainder     y^. 

79.  Divide  17,286  by  1,234.  Remainder     y^. 

80.  Divide  14,797,541  by  12,321.  Remainder  ^^i^. 

81.  A  man  bought  a  farm  for  14,400  dollars,  paying  75  dollars  au 
acre.     How  many  acres  did  it  contain  ?  A.   192  acres. 

82.  Suppose  a  farm  of  192  acres  cost  14,400  dollars,  how  much 
was  it  an  acre  1  A.  75  dollars. 

83.  In  1  pound  are  16  ounces ;  how  many  pounds  in  223,305 
ounces?  A.  1 3, 960/g^  pounds. 

84.  London  contained  in  1831  a  population  of  1,471,405.  Now 
allowing  that  13  persons  on  an  average  occupy  a  single  house,  how 
many  houses  then  would  be  required  to  accommodate  all  the  in- 
habitants. A.  113,185  houses. 

85.  How  much  is       yV    of  1,932,0451  ^.   148,618||. 

86.  How  much  is       4s    of  1,840,062?  A.     73,602i|. 

87.  How  much  is       ^^    of  2,500,368  ?  A.     55,563H. 

88.  How  much  is       ^^    of  4,210,9091  A.     67,683ff. 

89.  How  much  is       ^V    of  6,301,8951  A.     74,139f^. 

90.  How  much  is    S-^/^-M  A.       1,740^^. 

91.  Howmuchisi2|fll1  A.       l,835|f. 

92.  How  much  is    8-^-^1  A.       l,112ff. 

93.  According  to  the  census  of  1830,  the  entire  population  of  the 
U.  S,^  was  about  12,840,534,  and  the  number  of  children  who  never 
attend  school  is  about  4i  of  the  entire  population ;  what  was  their 
number  1  "  A.  611,454. 

XXIV.  When  the  divisor  is  a  composite  number. 

RULE. 

1.  Divide  first  by  one  factor  and  the  quotient  by  the  other. 

Q.  What  is  the  General  Rule?  70,  71.  In  what  cases  are  two  or  more 
figures  to  be  annexed  to  the  remainder  ?  72.  When  must  the  figure  you  place  m 
the  quotient  be  made  larger  or  smaller?  73.  What  is  the  order  of  proceeding? 
74.  Proof?  75. ______«^ 

1  u.  S.,  for  the  United  States  of  America. 


SIMPLE    DIVISION.  59 

2.  A  prize  of  7,200  dollars  is  to  be  divided  equally  among  36  men ; 
what  is  each  man's  part  1 

1  2  )  7  2  0  0  Had  there  been  but  12  men,  each  one's  part 

3  )       6  0  0         would  have  been  600  dollars ;  but  there  being 
2  0  0         3  times  as  many,  each  one's  part  is  only  ^  as 
'         much. 

3.  How  many  times  is  30  contained  in  1,230,  using  the  factors  3 
and  10?  A.  41. 

4.  Divide  1,152  by  24,  using  first  the  factors  4  and  6  ;  then  3  and 
8  ;  and  lastly,  2  and  12.  A.     48. 

5.  Divide  8,640  by    36.  A.  240. 

6.  Divide  2,160  by  144.  A.     15. 

7.  Divide  4,320  by    72.  A.     60. 

8.  Suppose  a  man  has  a  bundle  of  cloth,  in  which  are  4  pieces, 
containing  each  7  yards,  (making  28  yards  in  the  bundle;)  how  many 
such  bundles  would  270  yards  make  ? 

There  are  38  pieces  and  4  yards 
7)270  yards.  over,  and  9  bundles  and  2  pieces  over. 

4 )       3  8.  .  4  yards  over.       ^^  ^"^  ^ow  many  yards  would  remain 

9T72  pieces  over.       ^'^"^  ^^^i^H""^  1"^  ^^'  J^^^^iply  the  2 

=====  pieces  over  by  the  first  aivisor  [7  yards 

A.  9  bundles  18  yards,  or         in  each   piece,]    making   14    yards, 

9^1  bundles.  which  added  to  the  4  yards  over  at 

first,  makes  18  yards  remainder. 

9.  Hence  to  find  the  true  remainder  multiply  the  last  remainder  hy 
the  first  divisor  and  add  in  the  first  remainder. 

10.  Divide  85,509  by    42,  using  two  factors.  A.  2,035j#. 

11.  Divide  71,252  by    35,  using  two  factors.  A.  2,035f|. 

12.  Divide  81,605  by  108,  using  two  factors.  A.    755//?. 

13.  Divide  49,823  by  24,  using  its  three  factors,  2,  3  and  4; 
for  2  times  3   are  6,  and  4     ^^j^j  j        3    ^j^^  j^^^  remainder, 

times  6  are  24.  ^y      i  the  second  divisor, 

9 

Add  2,  the  second  remainder. 

Multiply  1  1 

by       2  the  first  divisor. 

Writing  under  the  23  the  Add     1  the  first  remainder 

total  divisor  makes  |f .  2~3  the  true  remainder. 

A.  2,075|f.  = 

14.  Divide  33,370  by  162,  using  the  factors  9,  6,  and  3  ;  for  they, 
multiplied  together,  make  the  total  divisor.  A.      205|-^-2-. 

15.  Divide  2,310,523  by  60,  using  3,  4,  and  5.       A.    38508^,^. 

16.  Divide  2,310,523  by  60,  using  2,  6,  and  5.       A.  38,508f^. 

XXIV.  Q.  What  is  the  rule  for  dividing  by  a  composite  number  ?  1.  How  is 
the  true  remainder  obtained  ?  9.  What  are  the  two  factors  of  48  ?  Of  108. 


)  4  9  8  2  3 

)  2  4  9  1   1. 

.  1 

4)8303. 

.2 

2  0  7  5. 

.3 

^  i  AniTiniETic. 

17.  Divide  2,310,523  by  60,  using  2,  2,  and  15.    A.  38,508||. 

18.  Hence  dividing  by  either  factor  firsts  brings  the  same  result. 

19.  Divide  3,707,716  by  120,  using  3  factors.      A.  30,897-^^0- 

20.  There  arc  1,728  solid  inches  in  1  foot;  how  many  solid  feet 
then  in  14,323  solid  inches,  using  12  as  a  factor  3  times.  A.  ^t^^s- 

21.  In  a  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high,  are 
128  solid  feet  or  1  cord  ;  how  many  cords  of  wood  then  in  a  pile  con- 
taining 41,726  solid  feet]  A.  325||f  cords. 

XXV.  When  the  divisor  is  10,  100,  1000,  &c. 

RULE. 

1.  Cut  off  as  many  figures  from  tlie  right-hand  of  the  dividend  as 
there  are  ciphers  in  the  divisor ;  the  figures  cut  off  are  the  remainder^ 
and  the  other  figures  of  the  dividend  the  quotient. 

2.  At  10  dollars  a  barrel,  how  many  barrels  of  flour  may  be 
purchased  for  369  dollars  1 

1  j  0  )  3  6  I  9  Removing  the  9,  makes  36    hundred 

A.   ~~3  6 1^0  barrels.         become  only,  36;   that  is,  tens  become 
units,  hundreds,  tens,  &c. 

3.  Divide  875,000,329  by    10.  A.  Remainder       ■^. 

4.  Divide  95,013,421  by    100.  A.  Remainder       f^. 

5.  Divide  8,732,509  by  1,000.  A.  Remainder     -j^V^. 

6.  Divide  340,137  by    10,000.  A.  Remainder    tww- 

7.  Divide  45.815  by    100,000.  A.  Remainder  AWA-  . 

8.  There  are  10  mills  in  every  cent ;  how  many  cents  then  ii 
895,430  mills'?  A.  89,543  cents. 

9.  Since  100  cents  make  1  dollar,  how  many  dollars  are  there  in 
9,503,200  cents.  A.  95,032  dollars. 

10.  How  many  dollars  in  8,534  cents,  and  how  many  cents  over  1 

A.  85  dollars  and  34  cents. 

11.  In  08,546,035  cents,  how  many  dollars  and  cents  1 

A.  685,460  dollars  and  35  cents. 

XXVI.  When  any  divisor  has  ciphers  on  the  right. 

RULE. 

1.  Cut  off  the  ciphers  from  the  right  of  the  divisor ^  and  an  equal 
number  of  figures  from  the  right  of  the  dividend. 

2.  Divide  the  remaining  figures  as  usual,  and  annex  to  the  right- 
hand  of  the  remainder  for  the  true  remainder,  all  the  figures  cut  off 
from  the  dividend. 

3.  Divide  89,952  dollars  equally  among  500  persons. 

XXV.  Q.  When  the  divisor  is  10,  100,  &c.  what  is  the  rule?  1.  What  is  the 
effect  of  removing  figures  from  the  right  of  the  dividend  ?  2.  What  is  y^  of  899  ? 
A.  89  9  .  What  is  ^  of  565  ?     »     of  836  ? 

XX  VI.  Q.  When  any  divisor  has  ciphers  on  the  right,  what  is  a  concise  rule? 
1  What  is  the  effect  of  cutting  off  two  figures?  3.  In  dividing  89,952  by  500, 
why  is  the  52  annexed  to  the  remainder  4  ?  3. 


CONTRACTION    OF    RULES.  61 

6l0  0  )  8  9  9  I  5  2  Cutting  off  two  figures  does  in  fact 

A.     I  1  9^ff.  divide  it  by  100,  it  being  one  factor  of  the 

•  composite  number  500.     Then,  if  we  mul- 

tiply (as  in  XXIV.  9,)  the  last  remainder  4  by  the  first  divisor  100,  it 
makes  400,  to  which  adding  52,  the  first  remainder,  makes  452  ;  but 
by  simply  annexing  the  52  to  4,  produces  the  same  effect,  hence  the 
rule. 

4.  Divide  783,456,078  by  2,100.  A.  Remainder  ^\. 

5.  Divide  634,278,975  by  8,000.  A.  Remainder  UU. 

6.  Divide  854,267  by  500,000.  A.  Quotient    Iff^f  S^. 

7.  The  annual  expense  for  schools  in  the  United  States  is  about 
15,000,000  of  dollars,  and  the  number  of  children  about  3,750,000  ; 
what  is  the  average  expense  for  each  child  1  A.  i  dollars. 

8.  The  number  of  teachers  is  about  95,000  :  how  many  scholars 
then  to  each  teacher  1  A.  40  nearly. 


CONTRACTION  OF  RULES. 

XXVII.  1.  To  multiply  easily  by  any  number  from  10  to  20. — 
Multiply  by  the  unit  figure  only  of  the  multiplier,  and  having  removed 
its  product  one  place  further  towards  the  right  of  the  multiplicandt 
add  it  to  the  multiplicand. 

2.  Multiply  8,978  by  19.  (3.)     8  9  7  8 

OPERATION.  1_9 

8978x19  8~0  802 

80802  8978 


170582  A.  170582  A. 


4.  For  the  figures  which  are  added  in  both  operations  are  the 
same ;  the  results  must  therefore  correspond. 

5.  Multiply  765,342,001  by  13.  A.     9,949,446,013. 

6.  Multiply  678,320,131  by  15.  A.  10,174,801,965. 

7.  Multiply  308,954,201  by  18.  A.     5,561,175,618. 

8.  Multiply  608,753,108  by  12.  A.     7,305,037,296. 

9.  Multiply  735,080,951  by  11.  A.     8,085,890,461. 

10.  To  multiply  by  5. — Annex  a  cipher,  and  divide  2. 

11.  For  annexing  one  cipher  multiplies  it  by  10,  then  2  times  5 
being  10,  dividing  only  by  2,  leaves  the  number  increased  5  times. 

12.  Multiply  6,545  by  5.  A.  32,725. 

13.  Multiply,  by  this  rule,  7,521  by  5.  A.  37,605. 

14.  To  divide  by  5. — Reverse  the  last  process  by  multiplying  by  2, 
and  cutting  off  one  figure  for  a  remainder. 

XXVII.  Q.  How  can  you  multiply  by  11, 12, 13,  &c.,  up  to  20  expeditiously? 
1.  Why  so?  4.  How  can  you  multiply  by  5  easily?  10.  Why?  11.  How  divide 
by  5  ^  14.  How  many  are  5  timea  48? — times  5  in  240? 
6 


62  ARITHMETIC. 

15.  Divide,  by  this  rule,  32,725  by  5.  A.  6,545. 

16.  Divide,  by  this  rule,  37,605  by  5.  A.  7,521. 

17.  To  multiply  by  25. — Annex  two  ciphers,  and  divide  by  4. 

18.  For  annexing  tvi'o  ciphers  multiplies  by  100,  and  4  times  25 
being  100,  dividing  only  by  4,  leaves  the  number  25  times  the  greater. 

19.  Multiply  6,532,405  by  25.  A.   163,310,125. 

20.  Multiply  4,230,216  by  25.  A.   105,755,400. 

21.  To  divide  by  25. — Reverse  the  last  process  by  multiplying  by  4, 
and  cutting  off  two  figures  for  a  remainder. 

22.  Divide  163,310,125  by  25.  A.  6,532,405. 

23.  Divide  105,755,400  by  25.  A.  4,230,216. 

24.  To  multiply  by  125. — Annex  three  ciphers,  and  divide  by  8. 

25.  For  8  times  125  being  1000,  annexing  three  ciphers  and 
dividing  only  by  8,  leaves  the  number  125  times  the  greater. 

26.  Multiply  6,304,521  by  125.  A.  788,065,125. 

27.  Multiply  2,403,450  by  125.  A.  300,431,250. 

28.  To  divide  by  125. — Reverse  the  last  process  by  multiplying  by 
8,  and  cutting  off  three  figures  from  the  right  of  the  product. 

'     29.  Divide  788,065,125  by  125.  A.  6,304,521. 

30.  Divide  300,431,250  by  125.  A.  2,403,450. 

31.  To  multiply  by  33  j. — Annex  two  ciphers,  and  divide  by  3. 

32.  For  3  times  33^  being  100,  annexing  two  ciphers  multiplies  by 
100,  and  dividing  only  by  3,  leaves  the  number  33j  times  the  greater. 

33.  Multiply  65,220  by  331.  A.  2,174,000. 

34.  Multiply  73,410  by  33-}.  A.  2,447,000. 

35.  To  divide  by  33}. — Reverse  the  last  process  by  multiplying  by 
3.  and  cutting  off  two  figures. 

36.  Divide  2,174,000  by  33}.  A.  65,220. 

37.  Divide  2,447,000  by  33}.  A.  73,410. 

38.  To  multiply  by  9,  or  99,  or  999,  &c. — Annex  to  the  multipli- 
cand as  many  ciphers  as  there  are  Qs,  and  subtract  the  multiplicand 
from  it. 

39.  For  annexing  one  cipher,  for  instance,  multiplies  by  10,  and 
deducting  the  multiplicand,  leaves  it  9  times  the  greater. 

40.  Multiply  467  by  9,  and  by  99,  and  by  9999. 
4670  46700  4670000 

4  6  7  4  6  7  4  6  7 

A.  4203  A.  46233  ^.4669533      ^ 


41.  Multiply  653,421  by  999.  A.  652,767,579. 

42.  Multiply  65,342  by  9,999.  A.  653,354,658. 


Q.  How  is  the  multiplying  by  25  abridged?  17.  Why?  18.  The  dividing  by 
25  abridged?  21.  How  many  are  25  times  8  ?  25  in  200?  How  is  the  multiplying 
by  125  abridged?  24.  Why?  25.  The  dividing  by  125?  28.  Multiply  8  by 
125.  Divide  1000 by  125.  How  is  the  multiplying  by  33}  abridged?  31.  Why? 
32.  The  dividing  by  33}  abridged?  35.  How  many  are  33}  times  15  ?— times 
33}  in  500?  What  abreviation  is  there  in  multiplying  by  9,  or  99,  or  999  ?  38. 
vAy  ?  39.  . 


ARITHMETICAL  SIGNS.  .  63 

43.  Multiply  6,534  by  99,999.  A.  653,393,466. 

44.  Multiply  653  by  999,999.  A.  652,999,347. 

45.  Multiply  65  by  9,999,999.  A.  649,999,935. 

46.  Multiply  6  by  99,999,999.  A.  599,999,994. 


ARITHMETICAL    SIGNS. 

XXVIII.  1.  Equality.^  The  sign  =  between  two  numbers  shows 
ihat  the  number  before  it  is  equal  in  value  to  the  number  after  it ;  as, 
100  cents  =  l  dollar,  meaning  100  cents  are  equal  to  I  dollar. 

2.  This  sign  is  two  horizontaP  lines,  drawn  parallel^  to  each  other. 

3.  Addition.*     The  sign + shows  that  the  number  before  it  is  to  be 

XXVIII.  Q.  What  is  the  sign  of  Equality  f  2.  What  does  it  show  ?  1.  What 
do  100  cents  and  1  dollar  with  the  sign  of  Equality  between  them  mean?    1. 

*rilOOF    OF   ADDITION,    SUBTRACTION,   MULTIPLICATION,    AND    DIVISION,  by  CflS^Z/l^ 

out  the  9s,  or  by  9s  as  it  is  called. 

*  Addition.  One  1 0  contains  one  9  and  1  unit ;  2  tens  or  20,  two  9s  and  2  units ;  3  tens 
or  30,  three  9s  and  3  units,  and  so  on,  leaving  for  a  remainder  each  lime  as  many  units  as 
there  are  tens. 

Hence  if  we  deduct  from  any  number  of  tens,  as  many  units  as  there  are  tens,  the 
remainder  will  contain  an  even  number  of  9s. 

One  hundred  [100  J  contains  eleven  9s  and  1  unit;  two  hundred  [200,]  twenty-two  93 
and  2  units,  and  so  on,  leaving  for  a  remainder  each  time  as  many  units  as  there  are 
hundreds. 

And  universally  if  from  any  number  of  tens,  or  hundreds,  or  thousands,  &c.,  there  be 
taken  as  many  units,  as  there  are  tens,  or  hundreds,  or  thousands,  Ace,  the  remainder 
will  contain  even  9s. 

The  number  634,  for  instance,  is  made  up  of  600,  30,  and  4.  The  COO  then  contains 
even  Os,  and  6  remainder ;  the  30,  even  9s,  ami  3  remainder ;  and  the  4  no  9s,  and  4 
remainder.  Now  the  remainders  6,  3,  and  4,  are  the  very  numbers  that  form  634,  there- 
fore we  derive  the  following  proposition,  (4)  on  which  is  based  (5)  the  proof  of  Addi- 
tion, viz  :— 

The  sum  of  the  figures  that  compose  any  number  has  an  excess  (6)  of  9s  equal  to  the 
excess  of  9s  in  that  number. 

Anfi  from  the  nature  of  Addition,  it  follows,  that  the  sum  of  the  excesses  of  9s  in  two 
or  more  numbers,  always  has  an  excess  equal  to  the  excess  in  tiie  sum  of  those  numbers. 
The  amount  is  1984,  and  the  proof  as  follows,  viz :  Adding  the  figures 


6  8  3 

9  5  6 

3  4  5 

19  8  4 


8  6,  8,  and  3  together,  in  the  top  line,  makes  17,  or  one  9  and  8  over  for 
2  the  excess ;  reject  the  one  9  and  write  down  on  the  right  the  excess 
^  above  9  which  is  the  8.  Do  the  same  with  the  9,  5,  and  6,  rejecting  the 
4  the  9s  and  v/riting  down  the  excess,  which  is  2.  The  third  line  leaves 
in  like  manner  an  excess  of  3  ;  next  adding  these  remainders,  3,  2,  and 


8,  makes  13,  or  one  9  and  4  remainder ;  reject  the  9  and  write  down  the  4  underneath. 
The  bottom  line  1,  9,  8,  and 4  makes  22,  or  two  9s  and  4  remainder  ;  rejecting  the  9s,  the 
remainder  is  4,  the  same  as  the  preceding  remainder,  and  therefore  1984  is  the  true 
amount. 

Rule.  Add  the  figures  in  the  uppermost  row  or  line  together,  reject  the  9s  contained  in 
their  sum,  and  set  the  excess  directly  even  ivith  the  figures  in  that  row.  Do  the  same  with 
each  row  and  set  all  the  excesses  of  9  together  in  a  line,  and  find  their  sum  ;  then  if  the 
excess  of  Qs  in  this  sum  (found  as  before,)  be  equal  to  the  excess  of  9s  in  the  total  sum,  the 
work  may  be  considered  correct. 

It  may  sometimes  be  more  convenient  to  reject  the  9s  while  adding  :  thus,  taking  5632 
for  example,  5  and  6  are  1 1— one  9  and  2  over  ;  the  2  over  and  3  are  5  and  2  are  8 

1  Equality,  fL.  egualitas.'i  Agreement;  evenness;  uniformity. 

2  Horizontal.  Relating  to  the  horizon  ;  level ;  not  perpendicular. 

3  Parallel.  A  line  equally  distant  through  its  whole  extent  from  another  line 

4  Proposition.  What  is  proposed  ;  statement  of  facts  ;  offer  of  terms. 

5  Hased.  Founded. 

6  Excess.  Wiiai  is  over ;  super.1iiity ;  remaining. 


64  ,  ARITHMETIC. 

added  to  the  number  after  it;  as  6  +  4  =  10,  meaning  G  and  4  added 
together  are  equal  to  10. 

4.  This  sign  is  a  cross,  formed  by  a  horizontal  line  intersecting*  a 
perpendicular^  one,  at  right^  angles,*  and  is  read  plus,^  which  means 
more;  thus  6+4  =  10,  means  6  plus  4  are  10. 

5.  How  many  are  375  +  125  +  100=  ^.600. 

6.  How  many  are  57,563  +  1,500  +  1,000,000  +  42  +  100  +  101+5 
+72?  A.  1,059,383. 

7.  Subtraction.*  This  sign— shows  that  the  number  after  it  is  to 
be  subtracted  from  the  number  before  it;  as,  6—4=2,  meaning  4 
from  6  leaves  2. 

8.  This  sign  is  a  single  horizontal  line,  and  is  often  called  minus,^ 
signifying  less;  thus,  10  —  3=7,  is  read  10  minus  3  is  7. 

9.  How  much  is  10,000,000-1,001  ?  A.     9,998,999. 

10.  How  much  is  37,500,209-4,209]  A.  37,496,000. 

1 1 .  Multiplication,  t  This  sign  x  shows  that  the  number  before 
it  and  the  number  after  it  are  to  be  multiplied  into  each  other ;  as, 
10x6=60,  meaning  10  times  6  are  60. 

12.  This  sign  is  two  lines  crossing  each  other  in  the  form  of  an  x 

13.  Howmany  are  5,320,065X801?  A.  4,261,372,065. 

14.  How  many  are  423x100x2001  A.  8,460,000. 

Q.  What  is  the  sign  of  Addition?  4.  How  formed?  4.  What  does  it  show  ?  3. 
How  is  it  read  ?  4.  How  much  is  6  plus  4  ?  8  plus  12  ?  What  is  the  sign  of  Sub- 
traction? 8.  What  is  it  often  called?  8.  What  does  it  show  ?  7.  How  much  is 
10  minus  7  ?  25  minus  14?  What  is  the  sign  of  Multiplication?  12.  What  does 
it  show  ?  1 1.  When  10  and  C  have  this  sign  between  them,  what  do  they  mean  ? 
11. 

*  SuBTEACTioN.  Since  the  subtrahend  and  difference  added  together  should  equal  the 
minuend,  the  Proof  is  the  same  in  principle,  as  that  for  Addition. 

Rule.     Reject  the  9s  from  the  subtrahend  and  difference,  noting  the  excesses.  Addthese 
excesses  together,  and  if  the  excess  of  9s  in  their  sum  iqual  the  excess  in  the  minuend,  the 
work  is  right. 
6  3  4  5  6... 6 

5  2  0  0  1  . . .  8  The  sum  of  the  excesses  8  and  7  are  15,  from  which  rejecting  the 

J   I  4  5  5        7        Os  leaves  6 ;  equaling  the  excess  in  the  minuend. 


t  Multiplication.    Since  Multiplication  is  an  abbreviation  of  Addition,  it  may  be 
proved  on  the  same  principle. 

Rule.     Multiply  the  excess  of  9s  in  the  multiplicand,  by  the  excess  in  the  multiplier,  and 
if  the  excess  of  9s  in  this  product  equal  the  excess  in  the  total  product  the  work  is  right. 
6  8  3  4  5.  . .8 
3  9.  . .3 

■ ^  The  product  of  the  two  excesses  Is  24,  and  the  excess  of  its 

«  «  i  n  o  ^  9s  is  6,  which  is  the  same  as  the  excess  in  the  product  and 

^  "  ^  "  ^  ^  therefore  right. 

2665455. ..6 


1  Inteksectino,  [L.  intersccto.l  Cutting  or  crossing  each  other. 

2  Peupendicular.  Hanging  in  a  straight  line  from  any  point  to  the  centre  of  the 
earth;  upright;  not  level. 

3  Right,  Straight ;  lawAil ;  just;  most  direct.  A  square  comer  is  a  right  angle;  a 
tjqnare  figure  has  four  right  angles ;  when  the  four  angles,  made  by  two  lines  crossing 
each  other  are  equal,  each  is  called  a  right  angle. 

4  Angle.  A  corner ;  the  space  between  two  lines  that  meet. 

5  Plus,  from  the  Latin  plus,  signifying  more. 

6  Minus,  from  the  Latin  minus,  sigiiifying  less. 


ARITHMETICAL    SIGx\S,  65 

15.  How  many  are  1x2x3x4x5x6x7x8x9x101 

A.  3,628,800. 

16.  Division.*  The  sign -^ shows  that  the  number  before  it  is  to 
be  divided  by  the  number  after  it ;  as,  CO  ^  5  =  12,  meaning  60  divided 
by  5  is  12. 

17.  The  sign  'j  shows  that  the  number  above  the  line,  is  to  be 
divided  by  the  number  below  the  line. 

18.  The  first  sign  is  formed  by  one  horizontal  line  passing  between 
two  dots,  and  the  second  by  writing  the  divisor  under  the  dividend 
with  a  line  between. 

19.  Perform   1,236,000^5.  A.  247,200. 

20.  Perform  3,756,000-^20.  A.   187,800. 

21.  Perform  ^-i^Y^/JiJLo ,  A.  8,458,085f|f 

22.  This  sign  is  the  proper  method  of  expressing  the  remainder 
after  division  is  performed,     vi.  1,  2. 

23.  Perform  750,348^125.  A.  6,002/TrV 

24.  Perform  32»-^'f  ij^a,  ^.  315,298^8^. 

25.  When  we  wish  merely  to  indicate  there  is  a  remainder,  it  being 
not  of  sufficient  importance  to  be  expressed,  the  sign  of  Addition  is 
generally  adopted  ;^  thus  10  mills -^3  makes  3  +  . 

26.  Divide  5,608,354  drams  by  117.  A.  47,934  + 

27.  Divide  7,503,478  gills  by  129  ?  A.  58,166  + 

28.  When  two  or  more  signs  occur  in  succession,  each  operation 
is  to  be  performed  in  the  order  of  the  signs. 

29.  Perform  600 +  100 -150X20 -Hi  1,000  =  l.t  A.  1. 

30.  Performl00  +  100-5  +  29^8+6x  11+40x3  +  617x5-1295 
-^80  =  100. A.  100. 

Q.  What  are  the  two  signs  of  Division  ?  18.  What  does  each  show  ?  16.  17. 
What  two  methods  are  there  of  indicating  a  remainder  ?  22.  25.  What  do 
several  signs  in  succession  indicate  ?   28. 

*  Division.  From  the  principles  of  proof  recognized  in  Addition  and  Multiplication, 
we  may  proceed  as  follows  : — 

85)38998(458  Divisor,          8  5 ,  ...  4  excess. 

3  4  0  Quotient,    4  5  8  ....  8  excess. 

4  9  9  ''     3  2  ,  ...  5  excess. 

4  2  5  Remainder,               fi  8 ,  .  .  .  5  excess. 

7  4  8  1  0  , ...  1  excess. 

6  6  0  Dividend,                   3  8  9  9  8,...!  excess. 


Rule.  Multiply  the  excess  of  9s  in  the  quotient,  btj  the  excess  in  the  divisor,  and  reject 
the  9s  from  the  product :  to  which  add  the  excess  of  9s  in  the  remainder,  and  if  the  result 
equal  the  excess  of  9s  in  the  dividend,  the  work  is  nght. 

Note.  This  property  of  9  belongs  to  it,  only  because  it  happens  to  be  1  less  than  10 
(the  radix  (2)  of  the  system ;)  for  did  we  reckon  by  Us,  then  10  would  answer  the  same 
purpose ;  but  since  any  number  of  9s  always  contains  an  exact  number  of  3s,  we  may 
prove  questions  as  well  by  casting  out  the  3s  in  the  manner  above,  as  by  casting  out 
the  9s. 

tAdd  100  to  600,  from  the  amount  subtract  150,  multiply  the  remainder  by  20,  and 
divide  the  product  by  11,000,  the  quotient  will  be  I  the  Answer. 

1.  Adopted,  [L.  adopto.'i  Taken  as  one's  own ;  selected  for  use. 

2  Radix,  [L.  radix,  a  root.'i  A  primitive  word ;  root. 
6* 


66  ARITHMETIC. 

31.  Perform  Vo"  + VV  X500-^250-15  =  5.* 

32.  Execute^  ^f^x  14-50-^4  +  ^^"  x  39^18 =563yV 


PROBLEMS. 

XXIX.  1.  A  Problem  is  a  question  proposed  for  solution;  in 
other  words,  it  is  something  to  be  done. 

2.  Prob.  I.  The  parts  of  a  number  being  given  to  find  that  num 
ber. — Add  the  several  parts  together. 

3.  If  the  several  parts  of  a  number  are  635 ;  4,008,  and  5,025,  what 
is  that  number  ?  A.  9,668. 

4.  The  total  value  of  real  estate  in  the  state  of  New  York,  in  1831, 
was  289,457,104  dollars,  and  of  personal  estate,  75,258,726  dollars; 
what  number  will  represent  the  value  of  both  1     A.  364,715,830. 

5.  Prob.  ii.  The  sum  of  two  numbers,  and  one  of  them  being 
given  to  find  the  other. — Subtract  the  given  number  from  the  given 
sum. 

6.  If  36,085  be  the  sum  of  two  numbers,  one  of  which  is  10,052, 
what  is  the  other  ?  A.  26,033. 

7.  What  is  that  number,  which,  with  25,233  will  make  36,085 1 

A.  10,852. 

8.  When  the  minuend  is  6,345  and  the  subtrahend  3,052,  what  is 
the  remainder  1  A.  3,293. 

9.  When  the  remainder  is  3,293  and  the  minuend  6,345,  what  is 
the  subtrahend.  A.  3,052. 

10.  Prob.  hi.  The  difference  between  two  numbersj  and  the 
greater  of  them  being  given  to  find  the  less. — Subtract  one  from  the 
other. 

11.  Whet  is  the  smaller  number  when  the  greater  one  is  15,675, 
and  their  difference  8,758]  A.  6,917. 

12.  When  the  remainder  is  4,080  and  the  minuend  6,304,  what  is 
the  subtrahend?  A.  2,224. 

13.  North  America  has  about  25,750,000  inhabitants,  and  the 
difference  between  the  population  of  North  and  South  America  is 
about  11,250,000 ;  how  many  inhabitants  has  South  America? 
A.  14,500,000. 

XXIX.  Q.  What  is  a  Problem?  1.  How  is  a  number  found  from  having  its 
parts  given?  2.  How,  from  having  one  of  two  numbers,  and  their  sum  given?  5. 
How  may  the  less  of  two  numbers  be  found,  from  having  the  greater  and  their 
difference  given?  10. 

*  Divide  100  by  50,  then  add  the  quotient  to  the  quotient  of  480  divided  by  60 ;  multiply 
that  sum  by  500 ;  divide  the  product  by  250  and  subtract  15  from  the  quotient  and  the 
remainder  will  be  5.  Answer. 

1  ExscuTE,  To  perform ;  to  finish ;  to  kill. 


PROBLEMS.  67 

14.  Prob.  IV.  The  difference  between  two  numbers,  and  the 
smaller  of  them  being  given,  to  find  the  greater. — Add  both  together. 

15.  If  the  difference  between  two  numbers  be  2,340,  and  the 
smaller  one  1,683,  what  is  the  greater  number  1  A.  4,023. 

16.  When  the  remainder  is  5,032  and  the  subtrahend  4,037,  what 
is  the  minuend  1  A.  9,069. 

17.  Suppose  a  man  who  was  born  A.  D.  1492,  the  year  in  which 
America  was  discovered,  to  live  as  long  as  Methuselah,  which  was 
969  years,  when  would  his  death  happen?  A.  D.  2,461. 

18.  France  has  a  population  of  about  32,000,000,  and  the  rest  of 
Europe  about  168,000,000 ;  what  then  is  the  entire  population  of 
Europe]  A.  200,000,000. 

19.  Prob.  v.  The  sum  of  two  or  more  numbers,  and  the  excess 
of  each  above  the  smallest  given,  to  find  those  numbers. — From  their 
whole  sum  subtract  the  given  excess,  and  if  there  be  more  than  two 
numbers,  subtract  the  sum  of  the  excesses  ;  then  divide  the  remainder 
into  2  equal  parts  to  find  two  numbers,  into  3  equal  parts  to  find  three, 
and  so  on ;  the  quotient  will  be  the  smallest  number ;  to  which,  add 
separately  the  several  excesses  at  first  subtracted  for  the  other  num- 
bers required. 

20.  What  are  the  two  numbers  whose  sum  is  1,824  and  the  excess 
of  the  one  above  the  other  360 1  A.  732  ;  1,092. 

21.  If  the  minuend  is  6,800  and  the  difference  between  the  remain- 
der and  subtrahend  be  850,  what  will  be  the  remainder  and  subtra- 
hend? A.  2,975;  3,825. 

22.  Two  men  having  met  on  a  journey,  found  by  calculation  that 
they  both  had  traveled  1000  miles,  and  that  one  had  traveled  150 
miles  more  than  the  other ;  what  distance  had  each  traveled  1 

A.  425;  575. 

23.  Suppose  that  two  fat  oxen  weigh  1950  pounds,  and  that  the 
difference  in  their  weight  is  215  pounds,  what  is  the  weight  of  each  1 

A.  867^  pounds ;  1,082|  pounds. 

24.  A  gentleman  gave  to  both  of  his  sons  65,300  dollars,  giving 

;h  receive  1 
A.  30,650;  34,650. 

25.  Suppose  that  John  has  done  in  one  day  20  sums  more  than 
Samuel,  and  Richard  30  sums  more  than  Samuel,  how  many  did  each 
ao,  allowing  all  to  have  done  260  ? 

Note. — Reckoning  what  John  and  Richard  both  did  more  than 
Samuel  makes  50  ;  the  sum  of  the  excesses  then  is  50  :  subtracting 
and  dividing  as  directed  in  the  rule  for  three  numbers  gives  70  for  the 
smallest  number,  that  is,  the  number  which  Samuel  did.  Adding  to 
70  John's  excess  over  that  makes  90,  and  adding  to  the  same  Richard's 
excess  makes  100. A.  S.  70:  J.  90;  R.  100. 

Q.  How  may  the  greater  of  two  numbers  be  found,  from  liaving  the  smaller 
and  their  difference  given  ?  14-  When  the  sura  of  two  or  more  numbers  and  their 
difference  are  given,  how  are  the  numbers  themselves  found  ?  19 


68  ARITHMETIC 

26.  Divide  16,000  dollars  so  that  B  may  have  300  more  than  A, 
and  C  400  more  than  A.         A.  A.  5,100;  B.  5,400;  C.  5,500. 

27.  Suppose  97  apples  are  so  divided  that  James  has  20  more  than 
Richard,  and  Thomas  30  more  than  James ;  how  many  has  each  1 

Note. — If  James  has  20  more  than  Richard,  and  Thomas  30  more 
than  James,  Thomas  evidently  has  50  more  than  Richard ;  and 
James  and  Thomas  together  have  70  more  than  Richard ;  the  sum 
of  the  excesses  here  then  is  70.  A.  R.  9  ;  J.  29  ;  T.  59. 

28.  A  father  has  an  estate  of  600,000  dollars  and  3  sons ;  he  gives 
to  the  second  son  50,000  dollars  more  than  to  the  youngest,  and  to 
the  oldest  50,000  more  than  to  the  second ;  what  sum  did  each 
receive  1 

Note. — The  oldest  has  50,000  more  than  the  second  son  and 
100,000  more  than  the  youngest ;  the  total  excess  then  is  150,000. 

A.   150,000;  200,000;  250,000. 

29.  Suppose  a  poor  man  has  labored  4  years  for  1,000  dollars, 
receiving  each  successive  year  50  dollars  advance  ;  what  sum  did  he 
receive  each  year  T 

Note. — The  2nd  year  he  received  50  more  than  for  the  1st.;  the 
3rd  year  100  more  ;  the  4th  year  150  more  ;  then  50  x  100+150= 
300,  the  excess ;     1000-300=700-^4  years  =  175. 

A.  175;  225;  275;  325. 

30.  Divide  1,300  dollars  so  that  B  may  have  300  more  than  A,  and 
C  200  less  than  A ;  what  sum  will  each  receive  1  The  excess  is 
100.  A.  400;  700;  200. 

31.  Prob.  VI.  The  sum  of  two  or  more  numbers  and  their  rate  of 
increase  or  decrease  being  given  to  find  those  numbers. — First  find 
what  sum  each  number  would  be  if  the  smallest  were  1 ;  ^then  add 
them  together  for  the  divisor  of  the  given  su7n,  the  quotient  will  be  the 
smallest  number;  with  which  proceed  as  with  the  1  at  first  to  find  the 
other  numbers  required. 

32.  Divide  900  dollars  so  that  B  may  have  3  times  as  much  as  A, 
and  C  4  times  as  much  as  B. 

Note. — Suppose  A  has  1,  then  B  will  have  (3  times  1=)  3  and  C 
(4  times  3  =  )  12.  Adding  1  and  3  and  12  together  makes  16  for  the 
divisor  of  1,600.  The  quotient  100  is  A's  part,  then  3  times  100= 
300  B's,  and  4  times  300  =  1,200  C's. 

A.  A's.   100  dollars;  B's.  300  dollars;  C's.   1,200  dollars. 

33.  A  man  bought  a  sheep,  a  cow,  and  a  horse,  for  165  dollars, 
paying  8  times  as  much  for  the  cow  as  for  the  sheep,  and  3  times  as 
much  for  the  horse  as  for  the  cow ;  what  price  did  he  pay  for  each  ? 
A.  5  dollars  for  the  sheep,  40  for  the  cow  and  120  for  the  horse. 

34.  Prob.  vii.  The  product  of  two  or  more  numbers,  and  one  of 

Q.  When  the  sum  of  two  or  more  numbers  and  their  rate  of  increase  are 
riven,  how  are  the  numbers  found  ?  31.  When  the  product  of  two  or  more  num 
bers,  and  one  of  them  are  given,  how  can  the  other  be  found?  34. 


PROBLEMS.  69 

them  being  given  to  find  the  other. — Divide  their  product  by  the  given 
number. 

35.  If  the  product  of  two  numbers  be  972  and  one  of  them  3C, 
what  is  the  other  1  A.  27. 

36.  If  375  be  a  product  and  15  a  quotient,  what  is  the  divisor? 

A.  25. 

37.  Suppose  one  of  the  factors  of  the  composite  number  972  is  27, 
what  is  the  other  factor  ?  A.  36. 

38.  Suppose  the  composite  number  11,250  has  factors,  the  product 
of  two  of  which  is  1,875,  what  is  the  third  factor?  A.  G. 

39.  When  18,000  barrels  of  flour  cost  234,000  dollars,  what  was 
the  cost  per  barrel  ?  A.  13  dollars. 

40.  Prob.  VIII.  When  the  quotient,  dividend  and  remainder  are 
given  to  find  the  divisor. — Divide  the  difference  between  the  dividend 
and  remainder  by  the  quotient. 

41.  Suppose  a  quotient  65,  a  dividend  9,520,  and  a  remainder  101 ; 
what  is  the  divisor  1  ^.145. 

42.  A  man  sold  a  block  of  buildings  containing  several  tenements 
and  shops  for  30,087  dollars,  being  on  an  average  for  each,  1,002 
dollars  and  27  dollars  besides.  How  many  shops  and  tenements 
were  there?  A.  30. 

43.  A  father  having  194,000  dollars,  gave  to  each  of  his  sons 
20,000  dollars  and  had  14,000  dollars  left.  How  many  sons  must  he 
have  had]  A.  9  sons. 

44.  Prob.  ix.  To  find  the  cost  of  several  things  from  having  the 
different  prices  of  each  given. — Add  the  different  prices  togther. 

45.  A  gentleman  purchased  a  farm  for  5,300  dollars,  a  vessel  for 
18,000  dollars,  a  span  of  fine  horses  for  950  dollars  ;  what  was  the 
cost  of  the  whole  ?  A.  24,310  dollars. 

46.  Prob.  x.  The  quantity  and  the  uniform  price  of  each  being 
given  to  find  the  cost. — Multiply  the  price  by  the  quantity. 

47.  What  is  the  cost  of  495,000  "  morus  multicaulis"  trees,  at  37^ 
cents  apiece?  A.  18,562,500  cents. 

48.  Prob.  xi.  The  cost  of  the  wiiole  and  the  equal  price  of  each 
being  given  to  find  the  quantity. — Divide  the  cost  of  the  whole  by  the 
price  of  one. 

49.  How  much  hay  will  1 ,836  dollars  purchase  at  12  dollars  a  load  ? 

A.  153  loads. 

50.  Prob.  xii.  The  quantity  and  cost  being  given  to  find  the  price 
of  one. — Divide  the  cost  by  the  quantity. 

Q.  How  is  the  divisor  found  from  having  the  quotient,  dividend,  and  remain- 
der given  ?  40.  How  is  the  cost  of  several  things  found  when  the  different  prices 
are  given  ?  44.  How,  when  the  price  is  uniform  ?  46.  How  is  the  quantity  found 
when  the  price  of  each  is  uniform?  48.  When  the  quantity  and  cost  are  given 
to  find  the  price  of  one?  50. 


*  TO  ARITHMETIC. 

51.  Wlien  a  cargo  of  8,500  bushels  of  wheat  sells  for  17,000  dol- 
lars, what  is  the  price  per  bushel  1  A.  2  dollars. 

52.  Prob.  XIII.  When  the  number  of  equal  parts  and  the  value  of 
one  are  given  to  find  the  value  of  the  whole. — Multiply  the  value  of 
one  by  the  ichole  number. 

53.  Suppose  a  packet  is  divided  into  8  equal  parts,  and  one  part 
sells  for  4,500  dollars,  what  is  the  whole  packet  worth  at  that  rate  \ 

A.  36,000  dollars. 

54.  If  I  (1-seventh)  of  a  beef  creature  cost  15  dollars,  what 
would  the  whole  cost  ?  A.   105  dollars. 

55.  There  are  105,192  seconds  in  \  of  a  year;  how  many  seconds 
then  in  a  year  ?  A.  525,960  seconds. 

66.  Prob.  xiv.  The  number  of  equal  parts,  and  their  value  being 
given,  to  find  the  value  of  one. — Divide  the  value  by  the  whole  num- 
ber of  equal  parts. 

57.  Suppose  the  capital  of  a  bank  to  be  200,000  dollars,  and  the 
number  of  shares  2,000 ;  how  much  is  each  share  ?  A.  100  dollars. 

58.  Suppose  a  packet  is  divided  into  eighths,  and  valued  at  36,000 
dollars ;  what  is  the  value  of  |  ]  A.  4,500  dollars. 


MISCELLANEOUS    EXAMPLES. 

XXX.  1.  When  one  of  two  numbers  is  3,750  and  their  sum  4,856, 
what  is  the  other.  A.  1,106. 

2.  When  the  remainder  is  one  million,  and  the  minuend  one  billion, 
what  is  the  subtrahend?  A.  999  milhon. 

3.  When  the  multiplicand  is  6,350  and  the  product  50,800,  what  is 
the  multiplier  ]  A.  S. 

4.  When  the  multiplier  is  8  and  the  product  50,800,  what  is  the 
multiplicand?  A.  6,350. 

5.  When  the  remainder  is  52,  the  quotient  49,  and  the  dividend 
17,937,  what  is  the  divisor  ?  A.  365. 

6.  When  the  remainder  is  ^y-^-^y  and  the  dividend  14,023,237, 
what  is  the  quotient  ?  A.  2003. 

7.  How  many  times  must  814  be  added  to  itself  to  make  407,000. 

A.  500  times. 

8.  How  many  times  must  500  be  taken  from  407,000,  to  find  how 
many  times  the  former  number  is  contained  ia  tlie  latter  ]  A.  814. 

9.  Suppose  a  farmer  has  his  live  stock  distributed  as  follows,  viz  : 
in  one  pasture  17  oxen,  7  calves,  and  4  young  horses ;  in  another  510 
sheep,  7  calves,  and  4  young  horses ;  in  his  yard  12  cows,  3  horses, 
5  young  oxen,  14  sheep,  10  colts,  17  turkeys,  15  hens,  25  geese,  and 
25  ducks ;  in  his  barn  22  hens,  14  calves,  11  geese,  and  3  ducks,  10 
sheep,  5  colts,  4  oxen,  3  cows,  and  2  horses.    What  is  the  amount 


MISCELLANEOUS    EXAMPLES.  71 

of  the  whole  1     A,  26  oxen,  28  horses,  15  cows,  28  calves,  534 
sheep,  17  turkeys,  36  geese,  28  ducks,  and  37  hens. 

10.  A  farmer  having  a  flock  of  700  sheep,  perceived  every  time  he 
foddered  them,  which  was  twice  a  day  for  30  days,  that  one  of  the 
number  was  missing ; '  how  many  sheep  had  he  left  at  the  end  of  the 
30  days  1  A.  640. 

11.  A  grocer  bought  12  barrels  of  flour  for  7  dollars  a  barrel,  and 
20  barrels  for  8  dollars  a  barrel.     What  did  the  whole  cost  1 

A.  244  dollars. 

12.  Suppose  a  man  who  had  500  dollars,  has  purchased  flour  to  the 
amount  of  244  dollars  ;  how  many  more  barrels  at  8  dollars  a  barrel, 
can  he  buy  with  the  remainder  I  A.  32  barrels. 

13.  Rufus  bought  a  vest  for  3  dollars,  a  hat  for  4  dollars,  and  for 
his  coat  he  paid  3  times  as  much  as  for  both  of  the  other  articles ; 
how  much  did  the  three  articles  cost  \  A.  28  dollars. 

14.  Suppose  a  poor  woman  has  only  230  cents,  with  which  to  pur- 
chase calico  for  a  gown,  and  suppose  that  it  takes  10  yards  for  a  pat- 
tern, how  high  a  price  can  she  pay  by  the  yard  to  just  take  all  the 
money  ?  A.  23  cents. 

15.  The  population  of  the  United  States  in  1830  was  about 
12,868,000.  Suppose,  as  has  been  computed,  1  person  to  every  400 
die  annually  by  intemperance,  how  many  deaths  then  in  the  United 
States  may  be  attributed  to  this  cause  alone  1  A.  32,170. 

16.  Suppose  only  30,000  die  annually  by  intemperance,  how  many 
is  that  a  month,  allowing  12  months  to  the  year  1  A.  2500. 

17.  What  would  be  the  expense  of  constructing  a  railroad  from 
Maine  to  the  Oregon  Territory,  the  distance  being  about  3,000  miles, 
at  the  cost  of  15,000  dollars  per  mile  1         A.  45,000,000  dollars. 

18.  How  long  a  time  would  it  require  to  travel  across  the  United 
States,  on  the  above  road,  at  the  rate  of  20  miles  an  hour,  or  480 
miles  a  day?  A.  O^f^  days. 

19.  Light  is  supposed  to  pass  from  the  sun  to  the  earth  in  about  8 
minutes,  a  distance  of  95  millions  of  miles.  How  far  then  does  light 
move  in  1  minute?  A.  11,875,000  miles. 

20.  The  national  debt  of  England  was,  in  1831,  about  3,300,000,000 
dollars,  and  the  revenue  of  Great  Britain  and  Ireland  about  300,000, 
000  annually  ;  suppose  this  revenue  to  be  applied  to  the  extinction  of 
the  debt,  how  long  a  time  would  it  require?  A.   11  years. 

21.  "  Mercury  is  the  smallest  and  swiftest  of  the  planets,  moving 
at  the  rate  of  111,000  miles  every  hour."  How  far  then  would  it 
move  in  24  hours,  or  1  day  ?  A.  2,664,000  miles.  How  far  in  305 
days,  or  1  year?  A.  972,360,000. 

22.  The  earth  is  about  25,000  miles  in  circumference  ;  how  many 
days  would  it  take  a  car,  at  the  rate  of  20  miles  an  hour  being  480 
miles  a  day,  to  move  round  the  earth  1  A.  52  jfYi)  <lays. 

23.  A  father  divided  his  property  so  that  his  sons  had  1,420  dollars 


72  ARITHMETIC. 

apiece,  and  his  daughters  1,100  dollars  apiece;  he  had?  sons  and  3 
daughters.    What  was  the  value  of  the  father's  estate  1 

A.  13,240  dollars. 

24.  A  flour  merchant  sold  108  barrels  of  flour  for  9  dollars  a  bar- 
rel, and  gained  on  it  2 16  dollars.  What  price  then  must  he  have  paid 
for  it  by  the  barrel  ?  A.  7  dollars. 

25.  A  purchased  of  B  500  bushels  of  wheat,  and  sold  it  for  1,475 
dollars,  which  was  475  dollars  more  than  its  cost.  What  must  he 
have  paid  a  bushel  for  it  ]  A.  2  dollars. 

26.  A  merchant  purchased  1,400  casks  of  lime  at  3  dollars  a  barrel, 
and  was  obliged  from  its  becoming  air  slacked,  to  sell  it  for  4,000  dol- 
lars.   What  was  his  loss  on  the  whole  1  A.  200  dollars. 

27.  How  much  can  a  cashier  lay  up  in  a  year  of  365  days  whose 
salary  is  1,500  dollars,  and  daily  expenses  3  doUars  1         A.  405. 

28.  A  farmer  sold  5  horses  which  cost  him  75  dollars  apiece,  for 
50  dollars  advance,^  and  received  payment  in  sheep  at  5  dollars  a 
head.     How  many  sheep  will  pay  for  the  horses  1      ^.85  sheep. 

29.  Suppose  a  certain  cistern,  which  will  hold  300  gallons,  has 
two  pipes ;  and  that  every  hour  25  gallons  ru^:  in  by  one  pipe,  and  15 
gallons  run  out  by  the  other,  how  many  gallons  would  stay  in  every 
hour  1  How  long  a  time  would  be  required  to  fiU  it  1 

A.   10  gallons;  30  hours. 

30.  Rufus  bought  a  watch  for  20  dollars,  and  paid  4  dollars  for  re- 
pairing it.  What  must  he  ask  for  it  to  gain  5  dollars  1  A.  29  dollars. 

31.  A  clock  strikes  1  time  for  1  o'clock,  2  times  for  2  o'clock,  3 
times  for  3  o'clock,  and  so  on  to  12  o'clock.  How  many  times  then 
will  a  clock  strike  in  half  a  day,  or  12  hours  1  A.  78  times. 

32.  If  a  clock  strikes  78  times  in  half  a  day  ;  how  many  times 
would  it  strike  in  a  whole  day  1  ^.  1 56  times.  How  many  times  in 
a  year  of  365  days  1  A.  56,940  times. 

33.  A  farmer  sold  a  grocer  10  bushels  of  corn  at  1  dollar  a  bushel; 
12  barrels  of  cider  at  2  dollars  a  barrel ;  he  received  in  payment  3 
barrels  of  flour  at  7  dollars  a  barrel,  and  the  balance  in  cash.  How 
much  money  did  he  receive  1  ^.13  dollars. 

34.  A  purchased  |  of  a  steamboat  for  1200  dollars ;  what  did  f 
cosf?  A.  2,400.  What  did  the  whole  boat  cost  ?  A.  6,000  dollars. 

35.  If  I  of  a  manufactory  be  sold  for  8,540  dollars,  what  would 
the  whole  bring  at  that  rate  1  A.  68,320  dollars. 

36.  How  many  times  can  ^  of  1,200  be  taken  from  1200,  and  have 
nothing  remain  1  A.  3  times. 

37.  Suppose  a  merchant  bought  200  barrels  of  pork  of  one  man, 
and  enough  of  another  man  to  make  750  barrels,  how  many  barrels 

»  did  he  buy  of  the  second  man?  A.  550  barrels. 

38.  A  farmer  has  475  bushels  of  grain  in  2  bins,  one  holding  125 
bushels  more  than  the  other ;  how  many  bushels  does  each  hold  1 
See  XXIX.  19.  A.  175  bushels;  300  bushels. 

1  Advance.  Moving  forward;  additional  price;  profit. 


FEDERAL    MONEY.  73 

39.  Suppose  two  persons  have  a  legacy  left  them  of  20,000  dol- 
lars, to  be  so  divided  that  one  may  have  3,000  more  than  the  other; 
what  is  each  one's  part  1  Give  one  3,000  dollars,  then  divide  the 
rest  equally.  A.  8,500  dollars;  11,500  dollars. 

40.  Suppose  a  man  traveled  from  Albany  to  Buffalo,  a  distance  of 
about  363  miles,  partly  by  the  canal  and  partly  by  the  railroad,  and 
that  he  went  163  miles  more  on  the  canal  than  on  the  railroad;  what 
distance  did  he  travel  on  each"?  A.   100  miles  ;  263  miles. 

41.  Divide  630  dollars  so  that  B  may  have  3  times  as  much  as  A, 
and  C  5  times  as  much  as  A.     See  xxix.  31.   A.  70 ;  210 ;  350. 

42.  Suppose  7  tons  of  hay  are  sufficient  to  keep  a  calf,  a  cow,  and 
a  horse,  through  the  winter,  and  that  the  horse  eats  2  times  as  much 
as  the  cow,  and  the  cow  2  times  as  much  as  the  calf;  what  quantity 
is  sufficient  for  each?  A.   1  ton  ;  2  tons ;  4  tons. 

43.  It  has  been  estimated,  that  the  population  of  the  globe  is  about 
816  millions,  and  that  every  32  years  as  many  inhabitants  as  are 
living  at  any  one  time  will  be  dead,  and  their  places  supplied  by 
others  ;  how  many  then  must  die  and  be  born  every  year  1 

A.  25,500,000  persons. 

44.  At  the  rate  of  25,500,000  persons  a  year ;  how  many  must 
die  and  be  born  every  day  of  the  year  (==365  days?)  A.  69,863  +  . 
How  many  every  hour  of  the  day  (==24  hours'?)  A.  2,911  nearly. 
How  many  every  minute  of  the  hour  (=60  minutes  ;)  or  in  less  time 
than  you  are  solving  the  question?  A.  48+. 


FEDERAL  MONEY. 

XXXI.  1.  Federal  Money  is  the  currency  or  coin  of  the  United 
States,  established  by  Congress  A.  D.  1786. 

2.  The  Eagle,  Dollar,  Dime,  Cent,  and  Mill,  are  the  several 
denominations  of  Federal  Money. 

3.  Accounts  are  kept  in  dollars  and  cents.  Eagles  and  dimes  are 
not  used  at  all,  the  former  being  expressed  in  dollars,  as  tens  of  dol- 
lars ;  and  the  latter  in  cents,  as  tens  of  cents. 

4.  Thus  5  eagles,  4  dollars,  6  dimes,  and  5  cents,  are  read,  54  dol- 
lars and  65  cents. 

5.  The  dollar  is  fixed  upon  as  the  unit  figure  ;  all  inferior  denom- 
inations are  therefore  parts  of  a  dollar. 

6.  Thus  the  dime  is  1-tenth  part  of  a  dollar,  the  cent  l-hundredth 
part,  and  the  mill  1 -thousandth  part.  

XXXI  Q.  What  is  Federal  Money?  1.  What  are  its  denominations?  2. 
Repeat  the  Table  of  Federal  Money  ?  See  vii.  1.  Which  denominations  are 
used  in  accounts  ?  3.  How  are  5  eagles,  4  dollars,  6  dimes,  and  5  cents  read  ?  4. 
How  is  the  value  of  the  different  denominations  determined  ?  5.  What  are  tha 
several  parts  of  the  inferior  denominations  ?  6. ^ 

1  Solving,  [L.  solvo.^  Loosing;  explaining;  unfolding;  performing. 
7 


74  ARITHMETIC. 


REDUCTION^  OF  FEDERAL  MONEY 

XXXII.  1.  Reduction'  is  the  changing  of  one  denomination  into 
another,  without  altering  its  value. 

2.  Thus,  2  dollars  into  200  cents  ;  40  mills  into  4  cents. 

3.  How  many  mills  are  there  in  8  cents  1 — in  27  cents  1 

4.  How  many  cents  in  80  mills  1 — in  270  mills  1 

5.  Reduce  2  dollars  to  dimes — to  cents — to  mills. 

6.  Reduce  2,000  mills  to  cents — to  dimes — to  dollars. 

7.  Reduce  5,000  cents  to  dollars — ^to  eagles.  ; 

8.  Reduce  5  eagles  to  dollars — to  cents. 

9.  Reduce  25  dollars  to  cents — 2,500  cents  to  dollars. 

10.  Reduce  34  dollars  to  dimes — to  cents — to  mills. 

11.  Reduce  34,000  mills  to  cents — to  dimes — to  dollars. 

12.  Reduce  815  cents  to  dollars  and  cents. 

13.  Reduce  8  dollars  and  15  cents  to  cents. 

14.  Hence,  annexing  two  ciphers  or  figures,  brings  dollars  into 
cents,  and  annexing  three,  into  mills,  and  vice  versa  ;^  that  is,  cutting 
off  two  figures  from  cents,  and  three  from  mills,  brings  them  back 
again  into  dollars. 

15.  Reduce  27  dollars  to  cents — to  mills. 

16.  Reduce  27,000  mills  to  cents — to  dollars. 

17.  Reduce  8  dollars  15  cents  8  mills  to  mills. 

18.  Reduce  8,158  mills  to  cents — to  dollars. 

19.  Dollars  and  cents  are  known  as  such  by  this  sign  $,  and  dis- 
tinguished one  from  the  other  by  a  point,  thence  called  a  Separatrix, 

20.  Thus  S8.356  means  8  dollars,  35  cents  and  6  mills. 

21.  Cents  then  occupy  the  first  two  places  on  the  right  of  dollars ^ 
and  mills  the  third  place. 

22.  Reduce  $7,156  to  mills. 

23.  Reduce  7, 156  mills  to  dollars,  cents  and  mills. 

24.  Hence,  merely  removing  the  separatrix  brings  dollars,  cents 
and  mills  into  mills,  and  vice  versa. 

25.  Reduce  $15,756  to  cents  and  mills. 

26.  Reduce  15,756  mills  into  dollars,  cents  and  mills. 

27.  As  cents  occupy  two  places,  recollect  when  the  cents  are  less 
then  10,  to  write  a  cipher  in  the  tens''  place,  or  place  of  dimes. 

28.  Reduce  5  dollars  and  6  cents  to  cents.  A.  506  cents. 

29.  Reduce  4  dollars  and  2  cents  to  cents.  A.  402  cents. 

XXXII.  Q.  What  is  Reduction  of  Federal  Money  ?  1.  How  are  dollars  re 
duced  to  cents  and  mills  and  the  reverse?  14.  How  many  cents  in  5  dollars T 
Dollars  in  500  cents  ?  Mills  in  5  dollars  ?  Dollars  in  5,000  mills  ?  How  are  do! 
lars  and  cents  distinguished  the  one  from  the  other?  19.  What  places  do  cents 
and  mills  occupy  ?  21.  What  is  the  direction  for  supplying  vacant  places?  27,31. 

1  Reduction,  from  re,  L.  hack,  and  duco,  L.  to  bring  or  lead ;  hence  it  literally  means 
to  bring  back ;  to  reduce  ;  to  subjugate. 

2  Vice  versa,  L.  The  terms  being  exchanged.  Thus,  the  generous  should  be  rich, 
and  vice  versa ;  that  is,  the  rich  shoiUd  be  generous. 


ADDITION    OF    FEDERAL    MONEY.  75 

30.  Express  3  dollars  7  cents  by  the  signs.  A.  $3.07. 

31.  When  there  are  no  cents  ivrite  two  ciphers  in  their  place. 

32.  Express  9  dollars  5  mills  by  the  signs.  A.  $9,005. 

33.  Express  8  dollars  7  mills  by  the  signs.  A.  8.007. 

34.  Reduce  6  dollars  and  3  mills  to  mills.  A.  6.003  mOls. 

35.  Since  Federal  Money  increases  in  a  tenfold  proportion,  like 
whole  numbers,  its  operations  are  performed  in  like  manner. 


B     8 

.  1  7 

1  3 

.0  5 

.  1  7 

.0  8  5 

.0  0  8 

2  1 

.4  8  3 

ADDITION   OF  FEDERAL   MONEY. 

RULE. 

XXXIII.  1.  Add  dollars  to  dollars,  cents  to  cents,  d^c,  as  in 
Simple  Addition,  placing  the  separatrix  directly  under  the  separatrix 
above. 

2.  Add  together  $8.17;  $13.05;  17ct.;  8ct.  5m.;  and  8m. 
Note. — d.  stands  for  dimes,  ct.  for  cents,  and 

m.  for  mills. 

Observe  the  cipher  before  the  &-  cents  and  5 
mills,  and  the  two  ciphers  before  the  8  mills.    See 
xxxii.  27.  31.  Thensay,  8and5are  13;  setdown 
•^'  3  and  carry  1,  adding  as  usual. 

3.  Add  together  $36.75;  $1.50;  $1.25,  and  $1.43.    A.  $40.93. 

4.  Addtogether$5.035,  $6,075,  $4,127,  $13,125.  A.  $28,362. 

6.  Add  together  $57,  $36.42,  $52.01,  $6.05  ;  8  cents  and  4  mills, 
and  9  mills.     See  the  first  example  No.  2.  A.  $151,573. 

6.  What  is  the  amount  of  8  dollars  and  5  cents,  $252  and  3  cents, 
$150  and  5  mills,  $17  and  8  mills.  A.  $427,093. 

7.  A  man  bought  a  chaise  for  $126. 18,  a  watch  for  $85  dollars  and 
6  cents  ;  a  coach  for  $850  ;  a  hat  for  $6  and  9  cents ;  a  whip  for  62 
cents  and  5  mills.  What  did  he  pay  for  the  whole]  A.  $1,067,955. 

8.  Find  the  sum  of  one  dollar  and  two  cents,  twenty-five  dollars 
and  two  mills,  $19.09,  three  dollars  and  three  mills.    A.  $48,115. 

9.  A  man  gave  an  eagle  for  a  coat,  five  dollars  and  five  dimes  for 
a  pair  of  boots,  six  dimes  and  six  mills  for  a  pair  of  gloves,  fifty-five 
mills  for  blacking  his  boots.  What  did  he  lay  out  in  alH  1  eagle  = 
$10;  5  dimes =50  cents;  55  mills =5  cents  5  mills.  A.  $16,161. 

10.  Add  together  $8,  $4,  8ct.  3m.,  75ct.  19m.,  19d.  7m.  3ct., 
$425  and  Im.,  5  eagles  and  5  mills.  A.  $489,795. 

11.  Find  the  sum  of  $135,  5ct.,  25ct.,  $18,  5m.,  80d.,  6  eagles, 
3m.,  18ct..,  $9,  2d.,  3d.  and  3m.  A.  $230,991. 

12.  Find  the  sum  of  $18|-,  $8f,  $5|,  $16f,  and  $25^ 

Q.  How  many  cents  are  8  dollars  2  cents?  15  dollars  10  cents?  Dollars  in 
802  cents?  in  1510  cents?  in  8176  cents?  in  1000  mills?  in  1234  mills?  How 
many  dollars,  cents  and  mills  be  reduced  to  mills,  and  the  reverse  ?  24.  How 
are  operations  in  Federal  Money  performed  ?  35. 

XXXIII.  Q.  What  is  the  rule  for  Addition  of  federal  Money  ?  1. 


76  ARITHMETIC. 

Note.— $1=25  ct.;  $|=50  ct.;  $f =75  ct.  A.  $74.75  or  $74|. 

13.  A  man  purchased  a  plough  for  $7|,  a  harrow  for  $4|,  a  load  of 
hay  for  $20.50,  a  yoke  of  oxen  for  $62^,  and  a  horse  for  $150i.  What 
did  the  whole  cost  him  1  A.  $245. 

14.  Find  the  sum  of  $50.37^  $65,  $54.20^,  18ct.,  9ct.,  45ct.,  8m. 
and  $17.65j,  (|ct.=5mills.)  "  ^.$187,963. 

15.  Suppose  a  traveler's  expenses,  at  various  times,  were  as  fol- 
lows, viz  :  $1.25,  $.085,  $2,007,  $6.53,  45ct.,  3m.  and  Oct.;  what 
was  the  total  amount  1  A.  $11.18. 

16.  Suppose  a  father  divides  an  estate  equally  between  his  two 
sons,  giving  one  $61,537.87^ ;  what  was  the  value  of  the  estate  ? 

A.  $123,075.75. 

17.  Suppose  1-third  of  a  load  of  hay  costs  $5.37|,  what  is  the  load 
worth  at  that  rate  ^  A.  $16,125. 

18.  Suppose  a  ship's  cargo  to  be  valued  at  $37,507.12^,  and  1- 
fourth  of  the  ship  itself  at  $15,000,  what  is  the  value  of  both  ship 
and  cargo]  A.  $97,567.12^. 


SUBTRACTION  OF  FEDERAL  MONEY. 

RULE. 
XXXIV.   1 .  Place  the  numbers  as  in  Addition^  and  subtract  as  in 
whole  numbers. 

2.  From  $65.83  take  $39.95  ;  from  $137  take  49  cents ;  from  $13 
take  5  mills. 

$65.83  $137.00  $12,000 

3  9.95  .4  9  .005 

J..  $25.88  A.  $136.51  ^.$11,995 


3.  Subtract  $15.43  from  $84.91.  A.  $69.48. 

4.  Subtract  $3,005  from  $5,650.  A.  $5,646,995. 

6.  A  gentleman  owing  $6,53 7. 50|,  paid  $2,549,655;  how  much 
remained  unpaid  ]  A.  $3,987.85. 

6.  From  $1,000,  take  35  cents  5  mills.  A.  $999,645. 

7.  From  $10  take  2  cents  5  mills.  A.       $9,975. 

8.  From  $3  take  62|  cents.  A.       $2,375. 

9.  From  $1,000  take  let.  and  Im.  A.  $999,989. 

10.  From  940  dollars  take  one  dime.  A.     $939.90. 

11.  From  1  dime  take  let.  and  Im.  A.         $.089. 

12.  If  a  man  owes  $3,500,625,  and  has  property  worth  $6,500 ; 
how  much  will  he  have  left  after  paying  his  debts  ?  A.  $2,999,375. 

13.  A  man  purchased  a  barrel  of  flour  for  $8.50,  and  paid  $3.87^, 
what  was  there  still  due  ?  A.  $4,625.  " 

14.  Subtract  5  dimes  from  50  cents.  A.  0. 

XXXIV.  Q.  "What  is  the  rule  for  Subtraction?  1. 


..itJLTIPLICATION    OF    FEDERAL    MONEY.  77 

15.  Subtract  10  eagles  from  $100.  A.  0. 

16.  Subtract  55  mills  from  5ct.  and  5  m.  A.  0. 

17.  Subtract  $10  from  10  eagles.  A,  $90. 

18.  If  a  ship's  cargo  is  valued  at  $20,875,  and  the  ship  at 
$3 1,675. 3 1|^,  how  much  is  the  ship  worth  more  than  the  cargo  1 

A,  $10,800,311. 

19.  Suppose  1-fourth  of  a  ship  is  worth  $7,500,  and  the  whole 
ship  at  $30,000,  what  is  3-fourths  of  the  ship  worth  1  ^..$22,500. 

20.  If  a  merchant  buys  a  lot  of  cotton  for  $0,500,875,  and  sells  it 
for  $8,215.12,  what  are  his  profits  1  A.  $1,714,245. 


MULTIPLICATION  OF  FEDERAL  MONEY. 

RULE. 

XXXV.   1  Multiply  as  in  whole  numbers,  and  call  so  many  figures 
cents  and  mills,  on  the  right  of  the  ■product,  as  there  are  figures  of 
cents  and  mills  in  the  multiplicand  or  multiplier. 
2.  What  will  7  yards  of  cloth  cost  at  $2.50  a  yard  \ 
^2.50  7  times  $2.50  is  the  same  as  $2.50,  written  down 

L.  7  times  and  added  together,  which  would  make 

$  1  7  .  5  0  A.     $17.50. 

(3.)  (4.)  (5.) 

Multiply    $65.01  $151,015  $375   76 

by  25  2    001  5.03 

A.   $1625.25  A.  $302, 181.  015 A.  $189,007.28 


6.  Multiply  $6,301.25  by  203.  A.  $1,279,153.75. 

7.  Multiply  $420,135  by  652.  A.     $273,928.02. 

8.  Multiply  $1,075. 08  by  750.  A.        $806310. 

9.  At  $2.50  a  head,  what  will  7,350  sheep  cost?      A.  $18375. 

10.  What  will  the  winding  of  26,750  balls  of  thread  amount  to  in 
dollars  and  cents,  at  3  mills  each  1  A.  $80.25. 

11.  Suppose  a  man  receive  37|  cents  apiece  for  making  285  pair 
of  slippers,  what  does  he  receive  for  the  whole  1        A.  $106,875. 

12.  A  merchant  bought  200  pieces  of  calico,  each  piece  containing 
40  yards,  for  29  cents  a  yard ;  what  did  he  pay  for  the  whole  ? 

A.  $2320. 

13.  What  will  17  hogsheads  of  vinegar,  each  containing  63  gallons, 
come  to  at  13  cents  a  gallon  \  A.  $139.23. 

14.  A  merchant  bought  15  barrels  of  flour  for  $150,  and  sold  it  for 
11 1  dollars  a  barrel;  what  did  he  make  on  the  whole?  A.  $22.50. 

15.  A  farmer  bought  500  sheep  for  $1100.50,  and  sold  them  for 
$2.75  apiece;  what  did  he  gain  on  the  whole ?  A.  $274.50. 

'  XXXV.  Q.  Rule  for  Multiplication?  1.  What  will  4  yards  of  cloth  cost  at  40 
cents  a  yard  ?  At  50  cents  a  yard?  How  many  dollars  will  buy  4  yards  of  broad 
cloth  at  $1.50  a  yard?  At  $2.50? 
7* 


?d  ARiTHMEtie. 

16.  A  drover  bought  cows  for  $27.50  ahead,  and  sold  them  fbf 
$30|-  a  head ;  what  profit  would  he  make  at  that  rate  on  buymg  and 
selling  324  cows  1     Multiply  324  by  the  profit  on  one.      A.  S891. 

17.  Suppose  a  man's  daily  income  is  $3.40,  and  his  daily  expendi- 
tures $2.16,  how  much  will  he  have  saved  at  the  year's  end,  or  in 
365  days  T  A.  $452.60. 

18.  Suppose  a  merchant  pays  his  clerk  $11.50  a  month,  and  $2  a 
week  for  his  board,  what  sum  does  the  clerk  cost  him  by  the  year  of 
12  months,  or  52  weeks  ]  A.  $242. 

19.  A  sold  B  15  bushels  of  wheat  at  $2.75  per  bushel,  and  l7 
bushels  of  rye  at  $1.12|abushd,  for  which  B.  gave  him  a  hogsheads 
(63  gallons)  of  molasses  at  35  cents  a  gallon,  and  the  balance  in  cash  ; 
how  much  cash  must  B.  pay  A.  Ai  $88.32^. 

20.  A  farmer  gave  $3,755  for  1-half  a  barrel  of  flour  ;  what  was 
the  price  by  the  barren  ^.$7.51. 

21.  When  1-third  of  an  estate  sells  for  $1,652;75,  what  is  the 
value  of  the  estate  1  A:  $4,958.25. 

22.  If  1-fourth  of  a  ship  cost  $1,650,  and  the  cargo  $18,251. 62^^ 
what  is  the  value  of  both  ship  and  cargo  1  A.  $24,851,625.  " 

23.  If  1-fifth  of  a  hogshead  of  molasses  cost  $5.25,  what  is  a  hogs 
headworthi     ^.$26.25.    What  are  17  hogsheads  worth  ? 

A.  $446.25. 

24.  A  received  a  legacy^  of  $1,150,125  ;  B  3  times  as  much,  and 
C  4  times  as  much  as  B  ;  what  sum  did  C  receive  1    A.  $13801^. 

25.  Suppose  a  bank  has  $20,000.50  in  specie*  deposited^  in  its 
vault,*  and  a  circulation^  of  15  times  that  amount,  what  is  the  amount 
of  its  bills  in  circulation  ?  *  ^.$300,007^. 

26.  Suppose  that  1-eighth  of  a  ship's  cargo  sells  for  $875,375,  and 
that  the  ship  itself  is  worth  3  times  as  much  as  the  entire  cargo ; 
what  must  be  the  value  of  the  ship  1  A.  $21,009 


DIVISION   OF  FEDERAL  MONEY. 

XXXVI,  1.  When  the  dividend  only  consists  of  Federal  Money. 

RULE. 

2.  Divide  as  in  simple  numbers ^  and  the  quotient  will  be  the  answet 
tn  the  lowest  denomination  of  the  dividend^  which  may  then  be  brought 
into  any  other  denomination  required. 

XXXVI.  Q.  When  the  dividend  is  Federal  Money,  what  is  the  rule  foi 
dividing?  2. 

1  Legacy.  A  bequest ;  money  or  property  left  by  Will. 

2  Specie.  Coin,  copper,  silver,  or  gold,  used  as  money. 

3  Deposited.  Laid  down  ,  lodged  in  the  care  of;  laid  aside. 

4  Vault.  A  cellar ;  an  arch;  a  cave ;  a  grave;  a  receptacle  used  by  banks  for  the 
safe  keeping  of  money. 

5  Circulation.  The  act  of  moving  round  or  in  a  circle ;  a  series  in  which  thing* 
return  to  the  same  state ;  currency 


E»tVlsl5N    OF    fEfiERAL    MONfiV.  t9 

3.  Divide  $17.50  equally  among  7  men. 

$       ct. 
y)  1  7.  5  0  I'or  $17.50  (by  xxxii.  24)  =  1750  cents-7= 

A.  $2.5  0         250  cents=$2.50,  the  ♦ame  as  in  the  operation. 

4.  Divide -f  202. 568  by  8.  A.  $25,321. 

5.  Divide  $728,065  by  23.  ^.$31,625. 

6.  What  is  the  price  of  1  pouild  of  sugar,  when  20  pounds  cost 
13  ?  When  8  pounds  cost  $1  i 

(50  (6.) 

$     ct.  $      ct. 

2  0)3.0  0  8)1.000 

$  .  1  5=^=15  cents  A.  $.12  5=12kents  A. 


7.  Hence  when  there  is  a  remainder  in  dividing  dollars  : — Bring 
them  into  cents  hy  annexing  two  ciphers,  and  if  there  he  still  a  re- 
mainder ;  into  mills  hy  annexing  another  cipher. 

8.  When  250  bushels  of  oats  cost  $80 ;  what  is  the  price  per 
bushel?  JLi  32  cents. 

9.  A  farmer  sold  30  bushels  of  potatoes  for  $5.85  :  what  was  that 
per  bushel  1  A.  19^  cents. 

10.  A  man  bought  3  hats  for  $10.40 ;  what  was  that  apiece  ? 

^^10    A.  n*  n  "^^  business  all  the  mills  under  5  are  rejected  \ 

^  but  5  mills  are  reckoned  as  ^  a  cent,  and  all 

6  +  .      Qver  5,  and  not  exceeding  10,  as  1  cent. 

11.  A  father  divided  an  estate  of  $30,000  equally  among  13  chil-^ 
dren  ;  what  was  each  one's  parf?  A.  $2j307.69.  + 

12.  Suppose  a  man's  salary  to  be  $3,650.60  a  year  of  365  daysj 
what  is  that  for  a  single  day  ]  A.  $10.  + 

13.  A  man  buys  36  pounds  of  sugar  for  110.50;  what  is  it  a 
pound  1  A.  $.29.+ 

14.  Sujppose  1,013  sheep  cost  $2,532.25,  what  were  they  apiece  t 

A.  ^2AQQ+hutcallit%2.50. 

15.  Suppose  you  buy  2  knives  for  40  cents,  what  must  you  ask 
apiece  for  them  to  gain  10  cents  1  A.  25  c6ntSi 

16.  A  grocer  bought  20  barrels  of  apples  for  $70  ;  what  must  he 
Bell  them  for  per  barrel,  to  gain  $10*  A.  $4. 

17.  Suppose  a  boy  gives  50  cents  for  3  bails  and.  looses  One  of 
them;  how  much  do  the  2  left  stand  him  in  apiece ?  A,  $j25i 

18.  Suppose  a  grocer  pays  $70  for  20  barrels  Of  apples^  and  aftef 
a  while  2  barrels  become  so  rotten  as  to  be  worthless ;  what  price 
perbarM  will  indemnify^  him?  (See  No.  10.)  A,  $3.89. -h 

How  do  you  proceed  with  the  remainder?  7.  How  are  mills  to  be  regarded  in 
the  quotient?  10. 

1  Indemnify.  To  save  from  harm  or  loss,  to  make  good  ;  to  reimburse  to  one  what 
he  has  lost. 


80  ARITHMETIC. 

19.  A  merchant  bought  5  barrels  of  flour  for  $47.50  ;  he  kept  one 
barrel  himself,  and  sold  the  rest  for  what  he  paid  for  the  whole  ;  what 
was  that  for  each  barren  A.  $11.87|. 

20.  A  father  and  his  3  sons  received  a  legacy  of  $30,000.  The 
father  had  ^  and  the  remainder  was  equally  divided  between  the  sons ; 
what  was  each  one's  part  1      A.  Father's  $15,000  ;  son's  $7,500. 

21.  A  boy  having  $1,  purchased  3  knives,  which  took  all  the  money 
he  had  except  25  cents ;  what  did  he  pay  apiece  for  them  1 

A.  25  cents. 

22.  A  man  having  $174.60  purchased  4  cows  ;  which  took  all  the 
money  he  had  except  $25 ;  what  did  he  pay  apiece  for  them  1 

A.  $37.40. 

XXXVII.  1 .  When  the  divisor  and  dividend  both  consist  of  Federal 

Money. 

RULE. 

2.  Reduce  the  divisor  and  dividend  to  the  lowest  denomination 
mentioned  in  either,  then  divide  as  in  simple  numbers. 

3.  How  many  hats  at  $1.50  apiece  may  be  bought  for  $9 1 
ots  cts 

1   e^  f\  \  n  (\  n  /  a  A         Reduce  $9  and  $1.50  each  to  cents ;  then 
there  will  be  as  many  hats  as  there  are  times 
^  ^  "  150  in  900. 

4.  How  many  barrels  of  flour  may  be  bought  for  $300,  at  $7.50  a 
barrelt  A.  40 barrels. 

5.  When  molasses  is  42  cents  by  the  gallon,  and  $26.46  by  the 
hogshead,  how  many  gallons  must  a  hogshead  hold  1  ^.  63  gallons. 

6.  A  certain  vessel  was  owned  equally  by  so  many  persons  that 
each  part  was  only  $1,250  ;  what  was  the  number  of  owners,  sup- 
posing the  vessel  cost  $62,5001  A.  50  owners. 

7.  If  a  farmer  received  $21  profit  from  each  cow,  and  from  the 
whole  $945,  how  many  cows  must  he  have  had  ?         A.  45  cows. 

8.  When  corn  is  $1,125  per  bushel,  how  many  bushels  maybe 
bought  for  $963  ?  Bring  $963  into  mills.  A.  856  bushels. 

9.  When  horse-keeping  costs  87^  cents  a  day,  how  long  might  a 
horse  be  kept  at  that  rate  for  $1,575  ?  A.   1800  days. 

10.  How  many  melons  will  $5  purchase,  at  25  cents  apiece  1 

A.  20  melons. 

11.  How  many  times  greater  then  in  value  is  $5,  than  25  cents  1 — 
than  2  cents  5  mills  1  ^.20  tunes ;  200  times. 

12.  How  many  times  greater  is  $73  than  40  cents  1  A.   182|^. 

13.  What  number  multiplied  by  8  mills,  will  make  a  product 
equal  in  value  to  $20,000  ^  A.  2,500,000. 

14.  Divide  $304  by  23  cents? A.      1321^- 

XXXVII.  Q.  How  do  you  proceed  when  the  divisor  and  dividend  are  both 
in  Federal  Money  ?  How  many  times  greater  in  value  is  8  dollars  than  8  cents  ?— 
than  8  mills? 


MISCELLANEOUS    EXAMPLES.  81 

15.  If  you  pay  813  a  ton  for  hay,  how  many  tons  can  you  buy  for 
S240.50'?  A.   18y^3^'V  tons. 

16.  If  a  district  school  receives  from  the  state  fund  ^45  being  at 
the  rate  of  $1.25  ahead  ;  what  is  the  number  of  scholars'!  A.  36. 

17.  A  merchant  invested  $34,500  in  cotton,  at  11|  cents  a  pound, 
and  sold  it  again  for  12|^  cents  a  pound ;  how  many  pounds  did  he 
purchase,  and  what  was  the  profit  on  the  whole  I 

Note. — First  find  the  quantity  bought  as  before,  which  multiply  by 
the  profit  on  a  single  pound.      A.  300,000  pounds  ;  profit  13,000. 

18.  A  gentleman  invested  $18,000  in  trees  of  the  "genuine  morus 
multicaulis"  species,  paying  for  each  30  cents,  and  afterwards  sold 
them  for  40  cents  apiece;  how  many  trees  did  he  purchase,  and  how 
much  did  he  gain  on  the  whole ?      A.  60,000  trees;  gain  $6,000. 


MISCELLANEOUS    EXAMPLES. 

XXXVIII.  Add  together  the  following  numbers. — 

(1.)  (2.) 

One  dollar  and  one  cent,  One  thousand  dollars. 

One  hundred  dollars.  One  cent  and  one  mill. 

Two  dollars  and  two  cents,  One  dollar  and  nine  mills. 

Twenty  cents  and  five  mills,  Ten  dimes  and  three  cents. 

Six  dollars  and  one  mill,  One  million  dollars,  one  mill, 
A.     $10  9,236 


A 

.$1,0 

0  1, 

0  0  2. 

0  5  1 

(4.) 
From  One  billion  dollars 
Take  One  cent  and  one  mill. 

1. 

$999, 

999 

,999. 

,989 

(3.) 
From  Eight  million  dollars 
Take  Forty-five  cents. 
A.  $7,999,999.55 

5.  Multiply  six  hundred  dollars  fifteen  cents  and  five  mills  by  sixty- 
two  hundred.  A.  $3,720,961. 

C.  If  1,050  be  one  factor  of  a  certain  number,  and  1,113  another 
factor,  v.lint  is  that  number.  A.   1,168,650. 

7.  If  it  costs  G  cents  for  a  boy  to  go  once  into  the  museum,  how 
many  times  could  he  go  in  for  648  ?  How  many  times  could  4  boys 
go  in  for  the  same  money  1  Only  ^  as  many  times  as  1  boy. 

A.  800 times;  200  times. 

8.  There  are  four  numbers  ;  the  first  is  215,  the  second  401,  the 
third  625,  and  the  fourth  as  much  as  all  tlie  other  three  lacking  200  ; 
what  is  the  sum  of  them  all!  A.  2,282. 

9.  Write  down  three  hundred  and  fifteen,  multiply  it  by  twenty- 
nine,  subtract  one  hundred  and  thirty-five  from  the  product,  and  if 
you  divide  the  remainder  by  nine,  the  quotient  will  be  1,000. 

10.  Perform  40,000-500^20x25  +  625-50,000. 


82  ARITHMETIC. 

11.  Perform  78x6- 168-^60x40 -200.  A.   =0. 

12.  Perform -^^fg^-".  A.  333,333|^. 

13.  What  is  |  of  100,  or  the  5th  part  of  100 1  A.  20. 

14.  What  is  the  7th  part  of  3,567,895?  A.  509,699f. 
15l  What  is  the  49th  part  of  1,374,952]  A.  280,60|f. 
16.  There  are  two  numbers,  the  greater  of  which  is  37  times  45, 

and  the  less  26  times  19  ;  what  is  their  sum,difFerence  and  product? 

A.  2,159;   1,171;  822,510. 
17    What  number  deducted  from  the  8th  part  of  200  will  leave  10  ? 

A.   15. 

18.  If  one  man  receive  $500  more  than  another,  and  both  receive 
$25,896.60,  what  sum  does  each  receive"! 

A.  $12,698.30;  $13,198.30. 

19.  Suppose  a  stage  goes  3  times  as  fast  as  a  footman,  and  a  rail- 
road car  4  times  as  fast  as  the  stage,  and  that  they  all  go  800  miles ; 
how  far  does  each  go  ?  A.  50  miles ;  150  miles;  600  miles. 

20.  Suppose  a  box  to  contain  275  Grammars  and  412  Arithmetics, 
what  will  they  come  to  at  33  cents  apiece  1  $226.71 

21.  A  farmer  sold  a  grocer  30  bushels  of  potatoes  for  27  cents  a 
bushel,  for  the  payment  of  which  he  took  a  keg  of  molasses,  con- 
taining 8  gallons,  at  45  cents  a  gallon,  and  the  balance  in  cash ;  how 
much  money  did  he  receive  ?  A.  $4.50. 

22.  A  exchanged  with  B  37  bushels  of  apples  at  45  cents  a 
bushel ;  4  bushels  of  onions  at  63  cents  a  bushel ;  200  bushels  of 
corn  at  $1.08,  for  55  yards  of  cloth  at  $2.14  a  yard,  and  for  the 
balance  he  was  to  receive  the  cash ;  what  was  A's  due  ? 

A.  $117.47 

23.  When  oats  are  50  cents  a  bushel,  and  sugar  10  cents  a  pound, 
how  many  pounds  of  sugar  will  4  bushels  of  oats  purchase  ? 

A.  20  pounds. 

24.  When  the  market  price  of  cheese  is  7  cents  a  pound,  and  that 
of  salt  $1.10  a  bushel,  how  many  bushels  of  salt  will  440  pounds  of 
cheese  purchase  ?  A.  28  bushels. 


QUESTIONS   INVOLVING   FRACTIONS* 

PREPARATORY    TO    THE   OPERATIONS   WITH   COMPOUND   NUMBERS. 

XXXIX.  1.  To  add  halves  and  quarters. — Call  every  2-quarters 
(as  I  and  j)  §  or  ^ ;  every  2-halves  (as  ^  and  |)  1  whole,  and  every 
^-quarters  (as  \  and  f )  1  whole. 

XXXIX.  Q.  How  are  halves  and  quarters  added  with  whole  numbers?  1. 
Why  do  \  and  |-  make  A?  See  the  Note  after  1.  How  many  whole  ones  in  4 
and  i  added  together?— in  \  and  |? — in  1  and  |? — in  A,  \  and  |-  ? — in  -1,  2,  11 

*  For  the  explanation  of  Fractions,  See  Part  First  from  vi.  to  vii.;  which  the  learner 
would  do  well  to  revise  before  he  commences  tliis  chapter. 


QUESTIONS    INVOLVING    FRACTIONS.  83 

Note. — The  f  is  called  I  because  it  means  2  of  4  equal  parts,  that 
is  I  of  them. 

2.  How  many  cents  will  |  of  a  cent,  f-  of  a  cent,  I  of  a  cent,  t  of 
a  cent,  and  |  of  a  cent  make  added  together  ?  il.  2f  cents. 

3.  A  man  spent  6}  cents  for  a  glass  of  sarsaparilla,  18f  cents  for 
a  water  melon,  12|  cents  for  an  inkstand,  37|  cents  for  a  book,  what 
did  all  these  articles  amount  to  ?  A.  75  cents. 

4.  Rufus  bought  a  ball  for  12|  cents,  a  penknife  for  87^  cents,  a 
writing-book  for  12|  cents,  an  Arithmetic  for  18^  cents,  and  a  pencil 
for  6^  cents ;  what  did  he  pay  for  the  wholel  A.  $1.37|. 

5.  Suppose  a  farmer  has  in  one  bin  320|  bushels  of  oats,  in  an- 
other 62  i  bushels,  in  another  49 f  bushels ;  how  many  bushels  has 
he  in  all  his  bins  ?  A.  432f  bushels. 

XL.  To  multiply  by  halves,  thirds,  quarters,  &c. 

1.  A  father  having  three  sons,  promised  the  youngest  2  cents,  the 
next  older  son  1  cent,  and  the  oldest  ^  a  cent  for  every  sum  each 
would  do  that  day.  They  did  40  sums  apiece.  How  many  cents 
then  must  he  pay  each '?  A.  SO  cents  ;  40  cents ;  20  cents. 

2.  To  multiply  by  2,  we  take  the  multiplicand  2  times;  by  1,  we 
take  it  1  time ;  and  by  ^  we  take  it  ^  a  time,  that  is,  the  half  of  it. 

3.  Hence  to  multiply  by  i  i  &c.— Divide  the  muUiphcand  by  the 
figure  below  the  line.  .„  ,      ,  i     . 

4.  For  the  smaller  the  multiplier,  the  smaller  will  be  the  product. 

4)3600  5.  What  will  3,600  yards  of  ribbon  cost,  at  6^- 

^i  cents  a  yard  1 

9  0  0  Note.— Take  1  of  3,600 =900,  then  multiply  by 

2  16  0  0  6  as  usual,  and  add  900  to  that  product. 

$2  2  5.0  0  A.  22500  ct.  =$225. 

6.  Multiply  2  rods  by  5^  yards  (  =  1  rod.)  A.  11  yards. 

7.  Multiply  26  rods  by  5 1  yards.  A.   143  yards. 

8.  Multiply  2  degrees  by  69|  miles  (  =  1  degree.)  A.  139  miles. 

9.  Multiply  360  degrees  by  69|  miles.  A.  25,020  miles. 

10.  Multiply  640  sq.  rods  by  30}  sq.  yards  (  =  1  sq.  rod.) 

A.    19,360  sq.  yd. 

11.  Multiply  133  sq.  rods  by  272}  sq.  feet.      A.  36,209}  sq.  ft. 

12.  Multiply  60  by  i-by  5l,-by  12^  A.  20  :  320  :  732. 

13.  Multiply  400  by  S^\,  by  36^.  -4.  3,220 :  14  402. 

14.  Multiply  100  years  by  365}  days.  A.      36,525  days. 

15.  To  multiply  by  f ,  f ,  &c.— Divide  by  the  figure  below  the  line 
and  multiply  the  quotient  by  the  upper  figure.  Or  multiply  first  and 
divide  afterwards,  especially  if  there  be  a  remainder. 

XL.  Q.  How  often  do  we  take  the  multiplicand  in  multiplying  by  2,  1  and 
L?  See  2.  What  then  is  the  rule  for  multiplying  by  i,  |,  &c.?  3.  What  is  the 
principle  ?  4.  How  is  3,600  multiplied  by  6}  ?  4. 


84  ARITHMETIC. 

16.  How  much  is  I  of  24?  A.  6.  |  of  24?  A.  18.  8  times  241 
A,  192.  How  much  then  is  8^  times  24 1 

^83       4^2'4       ar^'^iU  Divide  24  by  4  and  multiply  by  3; 

"i       *  LzJl  ^         or  as  the  result  is  the  same,  mul- 

tiply first  and  divide  afterwards  : 
making  18  to  be  added  to  192 
(=24x8.) 


rr  6  3 

1  9  2  3         4)7  2 

2  10  18  18 


17.  Multiply  460  by  6?,— by  16|.  A.        3,105  ;  7,636. 

18.  Multiply  504  by  9 i,— by  29||.  A.  4,851;  14,962yV 

19.  Multiply  370  by  7f ,— by  85||.  A.     2,795| ;  31,810. 

20.  How  much  does  8yV  times  1,000  exceed  S^times  1,000? 

A.  670. 

XLI.  To  divide  by  halves,  thirds,  quarters,  &c. 

1.  How  many  yards  of  tape  may  be  bought  for  4  cents,  at  2  cents 
a  yard  ? — at  1  cent  a  yard ! — at  i  of  a  cent  a  yard  ? — at  5-  of  a  cent  a 
yard  ?  A.  2  yd ;  4  yd ;  8  yd ;  16  yd. 

2.  Hence  to  divide  by  \,  ^,  &c. — Multiply  by  the  lower  figure. 

3.  For  the  smaller  the  divisor  the  greater  will  be  the  quotient. 

4.  How  many  bushels  of  oats  may  be  bought  for  $1  at  ^  of  a  dol- 
lar a  bushel  ?— for  $1,000  ?  A.  2  bushels  ;  2,000  bushels. 

5.  How  many  yards  of  calico  at  3^  of  a  dollar  a  yard,  may  be  bought 
for  $1  ?— for  $1,200  ?  A.  5  yards  ;  6,000  yards. 

6.  How  many  times  greater  in  value  is  $1,800  than  $|? 

A.   14,400. 

7.  Since  f  is  3  times  as  much  as  5- :  |,  4  times  as  much  as  |,  &c. 
therefore, — 

8.  To  divide  by  f ,  f ,  &c. — Multiply  by  the  lower  figure,  and  divide 
by  the  upper  one.  Or  divide  first  and  midtiply  afte?'ivards,  lohen  it 
can  be  done  without  a  remainder. 

9.  How  many  bushels  of  rye  may  be  bought  for  $10,  at  |  of  a  dol- 
lar a  bushel  ?— for  $2,400  ?  ($10x8-^5;  or  10-^5x8.) 

A.   16  bushels  ;  3,840  bushels. 

10.  When  rye  is  |  of  a  dollar  a  bushel,  how  many  bushels  may  be 
bought  for  $1.50?— for  $300?  A.  2  bushels  ;  400  bushels. 

11.  How  many  times  greater  in  value  is  $6  than  f  of  a  dollar? — 
is  $3,000  than  |  of  a  dollar?  A.  9  times ;  3,750  times. 

12.  Divide  20,000  by  o\  — by  tVf  — by  o^o  — by  AVo  — by  I- 
Answers.   133,333^;  26,666^^;  58,823||;  160,000;  25,000. 

13.  At  $1|  a  bushel,  how  many  bushels  of  wheat  may  be  bought 
for  3  dollars?— for  1,200  dollars? 

Q.  How  much  is  1  of  24  ?  3  of  24 ?  How  is  24  multiplied  by  8^  ?  See  16. 

XLI.  Q.  How  many  yards  of  cloth  at  2  dollars  a  yard,  may  be  bought  for  \  of 
a  dollar?  See  2.  For  ^  of  a  dollar  ?  See  2.  What  is  the  rule  ?  2.  What  is  "the 
principle?  3.  What  is  the  rule  for  dividing  by  ^^  |,  &c.?  8.  How  many  times  is 
I  contained  in  10  ?  See  9.  In  2,400  ?  See  9. 


QUESTIONS    INVOLVING    FRACTIONS.  85 

Note. — Since  2  halves  make  1  whole  there  are  6-halves  in  $3, 
for  $3x2  halves  =  6  halves.  Again  $1  x2  halves=2  halves  +  1-half 
(=4)  3-halves  :  which  is  the  divisor  of  the  6  halves  ;  thus  6-^3=2. 
So  $1,200X2  halves=2,400  halves  to  be  divided  by  3  halves  also. 

A.  2  bushels  ;  800  bushels. 

14.  How  many  fishhooks  at  2|  cents  apiece,  may  be  bought  for  9 
cents?  4-quarters  =  l  whole;  then  2  cents x4-quarters  —  8-quarters 
and  1-quarter  (  =  i)  more  make  9-quarters  the  divisor,  and  9  cents  X 
4-quarters=36-quarters,  the  dividend,  A.  4  fishhooks. 

15.  At  $2^  a  yard,  how  many  yards  may  be  bought  $9,625  1 
$  23- 

_^-fifths.  $  9  6  2  5^^^^^^  jg_  We  multiply  the  $2  by 

1  0-fifths.  ,    -  .  .   Q  ,   ^   r  „^,    '  5,  andaddinthe  l  =  ll-fifths, 

_J.-fifth.  1   1  H  8  1  2_^-fifths.  andthe$9,625by 5=48,125- 

IJ-fifths.  ^-  ^  ^  ^  ^  yards,  fifths. 

16.  Hence  to  divide  by  5|,  6},  &c. — Multiply  the  whole  number  of 
the  divisor  by  the  figure  below  the  line,  and  add  in  the  figure  above  the 
line  for  a  new  divisor. 

17.  Multiply  the  dividend  by  the  same  multiplier  for  a  new  dividend, 
then  divide  as  usual. 

18.  At  $4|  a  yard,  how  many  yards  of  cloth  may  be  bought  for 
$231  for  $4,600?  A.  5  yards  ;   1,000  yards. 

19.  What  is  the  quotient  of  63  divided  by  2|  ?  Say  2x8+5=21 
the  divisor  :  63  x  8  =  504,  the  dividend.  A.  24. 

20.  Divide  1,102  by  4f,— by  5^,— by  lOyV 

A.  228;  203;  106^^^. 

21.  There  are  5|  yards  in  every  rod ;  how  many  rods  then  in  33 
yards? — in  57  yards?  A.  6  rods;   10^,-  rods. 

22.  There  are  69|  statute  miles  in  every  degree ;  how  many  degrees 
in  400  miles  1  A.  5||f  degrees. 

23.  Plow  many  square  rods  in  24,200  square  yards,  there  being  30| 
square  yards  in  every  square  rod  ?  A.  800  square  yards. 

24.  Divide  1,090  square  feet  by  21'2\  square  feet.       A.  ^j^^q. 

25.  How  many  years  are  there  in  143,830  days  allowing  365|^  days 
to  the  year?  A.  393}^^^  years. 

26.  How  many  nails  are  there  in  38  inches,  allowing  2}  inches  to 
make  1  nail?  2j=9-quarters,  the  divisor;  and  38x4  =  152,  the 
dividend:  the  quotient  is  16|  nails,  but  since  the  remainder  is  al- 
ways like  the  dividend,  the  8  over  in  dividing  is  8-quarters  of  an  inch  ; 
then  8-quarters -T- 4  quarters  =  2  inches.     A.   16  nails  and  2  inches. 

27.  Hence  dividing  the  remainder,  by  the  multiplier  of  the  dividend; 
will  give  a  quotient  of  the  same  denomination  zvith  the  given  dividend. 

28.  Divide  12  yards  by  5^  yards  (  =  1  rod.)    A.  2  rods  1  yard. 

29.  Divide  376  yards  by  5^  yards  (  =  1  rod.)     ^.68  rods 2  yards. 

Q.  Ilovvmatiy  pair  of  boots  at  5i  dollars  a  pair,  may  be  bought  for  11  dollars? 
For  22  dollars  ?  For  33  dollars  ?    What  is  the  rule  for  dividing  thus  ?  16.  17. 


86  ARITHMETIC. 

30.  Divide  73,050  days  by  3651  days.       A.  200  years  8  days. 

XLII,  1 .  To  find  the  cost  of  articles  when  the  price  is  an  aliquot  part 
of  a  dollar.  See  vii.  81,  82,  86. — Divide  the  given  quantity  by  the 
number  of  aliquot  parts  ivhich  it  takes  of  the  price  to  make  a  dollar^ 
the  quotient  will  be  the  cost  in  dollars.   See  the  Table,  vii.  86. 

2.  At  50  cents  apiece  what  will  790  cubical  blocks  cost  T  50  ct.  = 
$^ ;  divide  by  2  because  every  2  costs  $1.  A.  395. 

3.  At  33|  cents  apiece,  what  will  591  inkstands  cost  1  33^  ct.= 
$g-;  divide  by  3  because  every  3  costs  $1.  A.  $197. 

4.  At  25  cents  apiece,  what  will  980  trees  cost?  25  ct.=$|-; 
divide  by  4,  because  every  4  cost  $1.  A.  1245. 

5.  At  16|  cents  apiece,  what  will  480  mellons  cost  1  16|  ct.  =$| ; 
divide  by  6  because  every  6  cost  $1.  A.  $80. 

6.  At  12|  cents  apiece,  what  will  1,080  books  cost  1  12|-  ct.  =$| ; 
divide  by  8  because  every  8  cost  il.  A.  $135. 

7.  At  %\  cents  apiece,  what  will  960  steel  pens  cost?  6}  ct.  ==$iV> 
divide  by  16  because  every  16  cost  $1.  A.  $60. 

8.  At  12|  cents  a  yard,  what  will  12  yards  of  ribbon  cost  ?  What 
1,565  yards^cost?  A.  $1.50. 

^  )  1  2  •  0  0  8  )  1  ,  5  6  5  .  0  0  0 

$1.50  $195,625 


9.  Hence  when  there  is  a  remainder,  annex  two  ciphers  for  cents  and 
one  for  mills. 

10.  What  will  be  the  cost  of  the  following  articles,  viz  : — 
410  bushels  of  rye  at  50  cents  a  bushel  ?  A.  $205. 
360  bushels  of  oats  at  33^  cents  a  bushel  ?  A.  $120. 
415  bushels  of  apples  at  25  cents  a  bushel?                 A.  $103.75. 
417  yards  of  calico  at  20  cents  a  yard  ?                        A.     $83.40. 
815  yards  of  shirting  at  16|  cents  a  yard  ?              A.  $135,833.  + 
489  quarts  of  cherries  at  12^  cents  a  quart?              A.  $61.12^. 
853  quarts  of  cranberries  at  10  cents  a  quart?           A.  $85.30. 
353  pounds  of  cheese  at  6}  cents  a  pound?                A.  $22.06.+ 
426  pounds  of  beef  at  5  cents  a  pound  ?                     A.  $21.30. 

1 1 .  At  $2. 50  or  $2^  per  yard ;  what  will  4  yards  of  broadcloth  cost? 
1,853  yards  cost?  Multiply  by  2i.  A.  $10  ;  $4,632.50. 

12.  What  will  be  the  cost  of  the  following  articles,  viz. — 
201  bushels  of  rye,  at  $2.12^  or  $2^  per  bushel  ? 

A.  $427. 12A. 
640  acres  of  land  at  $25.06|,  or  $25yV  per  acre  ?    A.  $16,040. 
315  barrels  of  flour,  at  $5.25  or  $5^  per  barrel?    A.   1,653.75. 
94 1  gallons  of  oil  at  $1.16f,  or  $li  per  gallon?  A.  $1,097.83.+ 

XLII.  Q.  What  is  meant  by  an  aliquot  part  ?  See  vii.  81.  Give  several  ex- 
amples. See  VII.  82,  83.  Repeat  the  Table  of  Aliquot  parts?  See  vii.  86. 
What  is  the  rule  for  finding  the  cost  when  the  price  is  an  aliquot  part  of  a  dollar  ? 
1.  What  will  be  the  cost  of  96  yards  of  calico  at  50  cents  a  yard?— at  33^ 
cents?— at  25  cents?— at  16|  cents  ?— at  \2\  cents ?— at  Q>\  cents  ? 


QUESTIONS     INVOLVING    FRACTIONS.  87 

13.  To  find  the  quantity  when  the  price  is  an  aliquot  part  of  a  dol- 
lar.— Reverse  the  last  rule,  that  is,  multiply  the  whole  cost  by  the 
number  of  aliquot  parts  which  it  takes  of  the  price  to  make  one  dollar. 

14.  At  12i  cents  a  pound,  how  many  pounds  of  sugar  may  be 
bought  for  SI  ?— for  $500!  IS^  ct.=S|,  then  $|  or  $1  will  buy  8 
pounds  and  $500  will  buy  8  times  500=4,000  pounds. 

A.  8  pounds  ;  4,000  pounds. 

15.  When  flaxseed  is  16|  cents  a  peck,  how  many  pecks  may  be 
boughtfor  $11— for  $7,118 1  A.  6  pecks;  42,708  pecks. 

16.  How  many  Madeira  trees  can  be  bought  for  $829,  when  the 
price  of  each  is  33^  cents]  A.  2,487  trees. 

17.  A  gentleman  invested  $1,000  in  goods  of  various  kinds;  what 
quantity  of  each  did  he  purchase,  taking  their  several  prices  from  the 
following  memoranda,  viz  : 

For  calico  $150,  at  12^  cents  per  yard. 

For  gingham  $116,  at  16f  cents  per  yard. 

For  French  calico  $50,  at  25  cts.  per  yard. 

For  silk  $200,  at  50  cents  per  yard. 

For  shoes  $100,  at  33  j  cents  a  pair. 

For  gloves  $50,  at  20  cents  a  pair. 

For  cotton  balls  $50,  at  5  cents  apiece. 

For  cotton  cloth  $48,  at  6^  cents  a  yard. 

The  balance,  or  what  remains  of  the  $1,000  after  deducting  the 
cost  of  the  above  articles,  he  laid  out  in  linen  at  40  cents  a  yard;  how 
many  yards  did  he  buy  ]  A.  590  yards. 


A. 

1,200  yards. 

A. 

696  yards. 

A. 

200  yards. 

A. 

400  yards. 

A. 

300  pairs. 

A. 

250  pairs. 

A. 

1,000   balls. 

A. 

768  yards. 

BILLS    OF    PARCELS. 

XLIII.  1.  Bills,  or  Bills  of  Parcels,  are  statements  of  goods 
bought  and  sold,  with  the  particulars  of  price  and  quantity,  as  in  the 

following  examples. ^ 

~(2^)  New  York,  July  26,  1838. 

Chauncey  Ackley,  Esq.,  .  ,-  ,     ^    .  ^ 

Bought  of  John  Smith, 

20  merino  sheep,  at  $6  per  head $ 

25  calves,  at  $2. 12^  per  head 

200  pounds  of  cheese,  at  6^  cents  per  pound 

36  bushels  of  oats,  at  27f  cents  per  bushel 

17  bushels  of  corn  at  75  cents  per  bushel 

$208.36^. 

November  15,  1839,  Received  payment, 

John  Smith. 


Q.  Suppose  there  is  a  remainder,  how  do  you  proceed?  9.  How  many  yards 
of  cloth  may  be  bought  for  10  dollars,  at  50  cents  a  yard  ?— at  33i  cents  ?— at  25 
cents  ?— at  20  cents  ?— at  16|  cents  ?— at  12i  cents  ?— at  6|-  cents  ?— at  5  cents  ^ 
What  is  the  rule?  13. 


88  ARITHMETIC, 


(3.)  Boston,  June  6,  1839. 

Mr.  George  Simpson, 

Bought  of  Rufus  Paywell, 

8barrelsof  cider,  at  $2.12|  a  barrel $ 

6  bushels  of  corn,  at  $1.16f  per  bushel 

$24.00 

August  8,  1839,  Received  payment, 
Rufus  Paywell. 

merchant's  bill. 
(4.)  Philadelphia,  January  1,  1827. 

Messrs.  Clark  &  Brothers, 

Bought  of  Peter  Rice, 

3,800  yards  of  calico,  at  17f  cents  a  yard $ 

■40  pieces  of  blue  broadcloth,  each  37  yards,     .  .  . 

at  $4.62|  per  yard 

400  yards  carpeting,  at  $1.18  per  yard 

200  pieces  of  nankin,  each  42  yards,  at  $.39,      .   . 

per  yard 

$11,267.50 
Received  payment  for  Peter  Rice, 

John  Stimpson. 


REDUCTION  OF  COMPOUND  NUMBERS. 

SEE   THE   TABLES   OF   MONEY,   WEIGHTS,   AND   MEASUKES.   VII. 

XLIV.   1.  How  many  feet  in  36  inches  1 — inches  in  3  feet? 

2.  How  many  shillings  are  there  in  £5 1 — pounds  in  100  shillings  ? 

3.  How  many  farthings  in  1,200  pence  1 — pence  in  4,800  farthings! 

4.  How  many  shillings  in  1,200  pence  1 — pounds  in  100  shillings  1 

5.  How  many  days  in  2  years  1 — hours  in  730  days  ] — minutes  in 
17,520  hours  1— seconds  in  1,051,200  minutes  ?  years  in  63,072,000 
seconds  1 

6.  Hence  pounds  must  be  multiplied  by  what  makes  a  pound,  shil- 
ling by  what  makes  a  shilling,  hours  by  what  makes  an  hour,  <Sfc. 

7.  This  process  is  called  Reduction  Descending,  because  num- 
bers by  it  are  carried  down  to  lower  denominations. 

XLIV.  Q.  How  many  shillings  in  2  pounds? — in  8  pounds?  Pence  in  5  shillings? 
Shillings  in  60  ppnce  ?  How  many  pence  in  20  farthings  ? — in  10  shillings  ? — in  48 
farthings?  What  is  Reduction  ?  XXXH.  What  are  Compound  Numbers  ?  ix. 
11.  Give  an  example,  ix.  12.  How  are  numbers  to  be  multiplied  in  Reduction? 
6.  How  divided?  8.  What  is  the  former  process  called,  and  why?  7.  What  is  the 
latter  called,  and  why?  9.  How  many  shillings  in  £5. 10s.?  Pounds  in  80  shillings  ? 
How  many  rods  in  5  furlongs  2  rods  ?  Feet  in  150  inches  ?  Yards  in  37  feet? 


REDUCTION    OF    COMPOUND    NUMBERS.  89 

8.  Shillings  too  must  be  divided  by  as  many  shillings  as  make  a 
pound ;  minutes,  by  as  many  minutes  as  make  an  hour,  dfc. 

9.  This  process  is  called  Reduction  Ascending,  because  numbers 
by  it  are  carried  up  to  higher  denominations, 

10.  Reduce  £2.  5s.  6|d.  to  shillings,  pence  and  farthings. 

£.     s.     d.    qr.  11.  Since  20s.  =£1 ;   12d.  =  ls;4qr. 

2.     5.     6.     3.  =  Id. ;  there  will  be  20  times  as  many 

2  0  s.  multiplier.  shillings  as  pounds ;  12  times  as  many 

4  0  s.  pence  as  shillings,  and  4  times  as  many 

5  s.  added  farthings  as  pence. 

4    5   s.  PROOF. 

1  2  d.  multiplier.  12.  How  many  pounds  in  2,187  far- 

6  4  0  d.  things? 

6  d.  added. 

5  4  6  d.  4  qr.  )  2  1  8  7  qr. 

4  qr.  multiplier.  1  2  d.  )  5  4  6  d.  3  qr. 

S  1  8  4  qr.  2  0  s.  )  4  5  s.  6~d.  3  qr. 

^ 3^ qr^ added.  ^^^  £  2.5  s.  6  d.  3  qr. 

2  1  8  7  qr.  Answer. 

13.  How  many  farthings  in  £4.  10s.  8d.  2qr.  ? 

14.  How  many  pounds  in  4,354  farthings  ? 

15.  How  many  quarters  in  2  T.  5  cwt.  3  qr  ? 

16.  How  many  tons  are  there  in  183  quarters  ^ 

17.  How  many  drams  in  3  T.  17  cwt.  3  qr.  17  lb.  8  oz.  5  dr.  ? 

18.  How  many  tons  are  there  in  1,994,885  drams  ? 

19.  How  many  seconds  in  1  Y.  51  d.  13  h.  35  m.  40  sec.  1 

20.  How  many  years  are  there  in  35,991,340  seconds'? 

21.  How  many  rods  in  18  m.  3  fur.  15  rods? 

22.  How  many  miles  are  there  in  5,895  rods  ? 

REDUCTION    DESCENDING    AND    ASCENDING. 

23.  Reduction  Descending,  is  reducing  numbers  from  higher  de- 
nominations to  lower  ones. 

RULE. 

24.  Multiply  the  highest  denomination  given,  by  as  many  of  the  next 
lower  as  make  one  of  that  higher,  adding  in  as  many  of  that  lower  as 
are  in  the  given  sum,  and  so  on. 

25.  Reduction  Ascending,  is  changing  numbers  from  lower  de- 
nominations to  higher  ones. 

RULE. 

26.  Divide  the  lowest  denomination  by  as  many  of  that  as  make  one 
of  the  next  higher  denomination.  Divide  that  quotient  in  the  same 
manner,  and  so  on,  the  last  quotient  luith  the  several  remainders  will  ■ 
form  the  answer. 

Q.  How  are  pounds  reduced  to  farthings  1  Farthings  to  pounds  ?  12.  Why 
multiply  in  one  case  by  20,  12  and  4,  and  in  the  other  case  divide  by  these  same 
numbers?  11.  What  is  Reduction  Descending  ?  23.  Rule?  Reduction  Ascend 
mg?  25   Rule?  26. 

8* 


00  Arithmetic. 

S7.  Reduction  Deseending  and  Ascending  prove  each  other. 

ENGLISH    MONEY. 

B8.  How  many  farthings  are  there  in  £0.  8s.  4d.  2qr.  1 

29.  How  many  pounds  are  there  in  6,162  farthings? 

SO.  How  many  farthings  are  there  in  £25.  9|d.  1  (^d.=2qr.) 

31.  How  many  pounds  are  there  in  24,038  farthings' 

32.  How  many  times  are  there  6  pence  in  414  pence  1 

33.  How  many  pence  are  there  in  69  sixpences  1 

34.  How  many  6  pences  are  in  £40,000 1 

35.  How  many  pounds  are  there  in  1,600,000  sixpences? 

36.  How  many  pounds  are  there  in  $445 1* 

37.  How  many  dollars  are  there  in  £133.  10s.  1 

38.  How  many  guineas  of  21s.  each,  in  £588 1 

39.  How  many  pounds  are  there  in  560  guineas? 

40.  How  many  French  crowns  at  6  shillings  and  8  pence  each,  are 
equal  in  value  to  1,161,600  farthings'?  (6s.  8d.=80d.) 

41.  How  many  farthings  in  3,630  French  crowns  ? 

42.  How  many  half-pence  are  there  in  £505.  3s.  ll|d.  1 

43.  How  many  pounds  are  there  in  242,495  half-pence  \ 

44.  How  many  4^d.  pieces  are  there  in  96  guineas  ] — (4|-=18qr.) 

45.  How  many  guineas  in  5,376  four-pence  half-penniesf 

46.  How  many  threepences  and  sixpences  in  one  ninepence,  of 
each  an  equal  number]  6  +  3=9,  then  9-^9  =  1  of  each,  A. 

47.  Suppose  630  pence  to  contain  an  equal  number  of  sixpences 
and  3  pences,  what  is  that  number  T  A.  70. 

48.  In  £101.  5s.  how  many  guineas  and  dollars,  of  each  an  equal 
number?  '  A.  75. 

49.  Reduce  £32.  16s.  to  dollars  of  8sv  each.  A.  $82. 

50.  Reduce  £11.  5s.  to  dollars  of  7s.  6d.  each.  A.  $30. 

51.  Reduce  $360  of  4s.  6d.  sterling  to  pounds.  A,  £81. 

52.  How  many  dollars  will  purchase  7,200  bushels  of  potatoes,  at 
Is.  6d.  per  bushel? 

53.  How  many  bushels  of  potatoes  may  be  bought  for  $1,800,  at 
Is.  6d.  per  bushel? 

54.  When  7,200  bushels  of  potatoes  cost  $1,800,  what  is  their 
price  by  the  bushel  1 

55.  Suppose  a  merchant  imports  flrom  England  80,640  yards,  of 
tape,  at  |d.  per  yard,  how  many  pounds  will  pay  for  it  ?  A  £168. 

Q.  Proof  of  both?  27-.  What  is  English  Money?  vii.  2.  Repeat  the  Table. 
How  are  po\itids  brought  into  sixpences  ?  34.  Dollai-s  into  pounds  and  the  reverse  ? 
36.  Pounds  and  shillings  into  ^incas  ?  How  many  guineas  in  £2  and  2  shillings? 
Pounds  in 2  guineas?  Pounds  m  $6  and  4  shillings  ?  Dollars  in  £2  ?  How  many 
6  pences  in  3s.  Gd.  ?  How  are  farthings  brought  into  French  crowns  of  6s.  Sd. 
each?  40.  Guineas  into  four-pence  half-pennies?  45.  How  are  pounds  brought 
into  an  equal  number  of  3  pences  and  6  pences  ?  How  many  6  pences  and  9 
pences  in  2s.  6d.  ? — in  5s.  ? 

*  Allow  6  shillings  to  the  dollar  when  no  nuo*''    's  mentioned. 


RfilDUCTION    OF    WEIGHTS    AND    MEASURES.  9l 

TROY    WEIGHT. 

56.  How  many  pennyweights  in  24  lb.  3  oz.  15  dwt.  ? 

57.  How  many  pounds  are  there  in  5,835  penny  weights  T 

58.  What  is  the  value  of  a  silver  tankard,  weighing  41b.  8  oz.,  at 
$1.15  per  ounce  1 

59.  When  a  silver  tankard  costs  $64.40,  at  $1.15  per  ounce,  what 
will  be  its  weight  in  pounds  I 

60.  What  would  be  the  value  of  the  same  tankard  at  6  cents  a 
pennyweight?  ^'  $67.20. 

AVOIRDUPOIS    WEIGHTS. 

61.  How  many  ounces  in  5  cwt.  3  qr.  17  lb.  11  oz.1 

62.  How  many  hundred  weight  in  9,483  ounces? 

63.  How  many  drams  are  there  in  3  qr.  15  lb.  10  dr.  ? 

64.  How  many  quarters  are  there  in  23,050  drams? 

65.  In  14  hogsheads  of  sugar,  each  weighing  10  cwt.  14  lb.,  how 
many  pounds?  ^-   14,196  pounds. 

66  What  will  11  cwt.  2  qr.  15  lb.  of  sugar  cost,  at  12|  cents  per 
pound?  ^-  $145,621 

67.  How  many  small  boxes,  each  to  contain  25  lb.  may  be  filled 
from  85  hogsheads  of  tobacco,  each  weighing  8  cwt.  15  lbs. 

A.  2,771  boxes. 

apothecaries'    WEIGHT. 

68.  How  many  grains  in  5  ft  5  f  1  3  2  3  13  gr? 

69.  How  many  pounds  are  there  in  31,313  grains? 

70.  How  much  calomel  and  aloes  are  contained  in  36  boxes  of 
pills,  each  box  having  20  pills,  and  each  pill  2  grains  of  calomel  and 
8  grains  of  aloes  ?  (2  gr.  +  8  gr.  =  10  gr.)  A.  Ife  3f. 

71.  An  apothecary  having  mixed  in  proper  proportion  3  ounces  of 
calomel  with  1  pound  of  aloes,  wishes  to  find  how  many  boxes,  each 
to  contain  20  pills,  and  each  pill  2  grains  of  calomel  and  8  grains  of 
aloes,  will  hold  the  compound  ?  A.  36  boxes. 

CLOTH    MEASURE. 

72.  How  many  nails  are  there  in  750  yds.  1  qr.  1  na.  ? 

73.  How  many  yards  are  there  in  12,005  nails? 

74.  How  many  flemish  ells  are  there  in  1,080  yards?  A.   1,440. 

75.  How  many  quarters  are  there  in  6  pieces,  each  containing  20 
yards  and  1-quarter  ?        ^.  486  quarters. 

Q.  For  what  is  Troy  Weight  used?  vii.  5.  Repeat  the  Table.  How  are 
pounds,  ounces,  &c.  brought  into  grains?  How  are  drams  brought  into  tons  ' 
How  many  pounds  in  36  ounces  ?— in  40  oz.  ?  Ounces  m  3  pounds  ?— m  33_  lb.  ( 
For  what  is  Avoirdupois  Weight  used?  Repeat  the  Table.  How  many  pounds 
were  formerly  reckoned  for  a  quarter?  (See  reference  2,  at  the  bottom  of  p  12.) 
How  many  hundred  weight  in  2  tons  ?— m  2^  tons  ?  Pounds  m  2  quarters  ?— m 
5 Jj  quarters?  For  what  is  Apothecaries' "Weight  used?  Repeat  the  Table. 
How  many  scruples  in  60  grains  ?  Ounces  in  5  pounds  ?— in  ll|i  pounds  ?  What 
is  the  use  of  Cloth  Measure  ?  Repeat  the  Table.  How  many  yards  in  9  qr.  ? 
Quarters  in  lOf  yds  ?  How  are  French  ells  brought  into  nails?  Quarters  mto 
yards  ?  Yards  into  elis  Flemish  ? 


02  ARITHMETIC. 

76.  How  many  yards  of  cloth  only  2-quarters  wide,  is  equal  to  10 
yards  which  is  4-quarters  wide  1  A.  20  yards. 

77.  How  many  yards  1-qr.  wide  are  equivalent  to  50  yards  4 
quarters  wide  1  A.  200  yards. 

78.  What  will  200  yd.  2  qr.  of  cloth  cost  at  26  cents  per  quarter  1 
A.  $200^.     At  12^  cents  per  quarter  ?  A.  $100.25. 

DRY    MEASURE. 

79.  In  19,691  quarts  how  many  bushels'? 

80.  In  615  bus.  1  pk.  3  qt.  how  many  quarts'? 

81.  At  40  cents  a  peck  what  will  25  bushels  3  pecks  of  wheat 
cost?  A.  $41.20. 

82.  When  rye  sells  for  20  cents  a  peck,  how  many  bushels  maybe 
bought  for  $247.20  ?  A.  309  bushels. 

83.  How  many  bags  will  8,500  bushels  of  rye  fill,  allowing  each 
bag  to  hold  4  bushels  1  peck  '?  A.  2,000  bags. 

WINE    MEASURE. 

84.  How  many  pints  in  2  hhd.  40  gal.  3  qt.  1  pt,  1 

85.  How  many  hogsheads  are  there  in  1,335  pints  '? 

86.  A  tee-totaler  found  to  his  sorrow,  that  he  had  drank,  in  all  his 
life,  no  less  than  1  tun  of  wine  ;  what  would  it  have  amounted  to  at 
6^  cents  a  half  gill  ?  A.  $1,008. 

87.  A  merchant  bought  5  hogsheads  of  molasses  for  12^  cents  a 
quart,  and  sold  it  for  6|  cents  a  pint ;  did  he  make  or  loose  ? 

A.  Neither. 

ALE,    OR    BEER    MEASURE. 

88.  How  many  pints  are  there  in  1  hhd. '?  A.  432  pints. 

89.  What  will  45  bar.  18  gal.  of  ale  come  to  at  4  cents  a  pint  T 

90.  A  man  having  retailed  45  bar.  18  gal.  of  ale,  received  for  the 
whole  $524.16  ;  what  did  he  get  by  the  pinf? 

91.  How  many  l|  pint  bottles  can  be  filled  with  3  hogsheads  of 
ale^  A.  804 

Long  measure. 

92.  How  many  inches  in  100  yd.  2  ft.  5  in.  ■? 

93.  How  many  yards  are  there  in  3,629  inches'? 

94.  How  many  inches  in  1  mile '?  A.  63,360. 

95.  How  many  barley  corns  is  it  round  the  globe,  it  being  360 
degrees  t  See  XL.  3. 

Q.  When  cloth  is  only  2-quartcrs  wide,  bow  can  you  find  what  quantity 
4-quarters  wide  will  equal  it  ?  76,  In  15  yards  how  many  Flemish  ells  ?— how 
many  English  ells  ?  What  is  the  application  of  Dry  Measure  ?  Repeat  the  Table. 
How  are  bushels  reduced  to  pints?  Quarts  to  chaldrons?  How  many  pints  in  3 
quarts  ?  Quarts  in  10|  pk.?  (1  qt.  =  |  pk.)  Pecks  in  123  bushels?  What  is  the 
use  of  Wine  Measure  ?  Repeat  the  Table.  How  are  gills  reduced  to  hogsheads? 
Pipes  to  gallons?  How  many  gallons  in  24  quarts?— in  1^  hhd.?  For  what  is 
Ale  or  Beer  Measure  used  ?  Repeat  the  Table.  What  would  a  firkin  of  ale  cost 
at  50  cents  a  gallon?  What  would  a  firkin  of  beer  cost  at  the  same  price?  For 
what  is  Long  Measure  used?  Repeat  the  Table. 


REDUCTION    OF    WEIGHTS    AND    MEASURES.  93 

96.  Howmany  degrees  in  4,755,80 1,600 barley  corns'? 

07.  How  many  inches  is  it  from  Boston  to  Providence,  it  being  40 
miles?  ^.  2,534,400  inches. 

98.  Suppose  5  paces  to  make  1  rod,  how  many  paces  will  reach 
round  the  earth  1  ^.  40,032,000  paces. 

LAND    OR    SQUARE    MEASURE. 

99.  How  many  square  rods  in  5  square  miles  ? 

100.  How  many  square  miles  in  512,000  sq.  rods'? 

101.  In  60  sq.  m.  37  A.  17  R.  how  many  sq.  poles? 

102.  In  6,150,600  sq.  poles  how  many  square  miles'? 

103.  How  many  square  feet  in  a  room  15  feet  long  and  13  feet 
v^^ide  1  (For  the  rule  see  vii.  46.)  A.  195  sq.  ft. 

104.  How  many  square  rods  are  there  in  a  piece  of  land  120  rods 
long  and  17  rods  widel  A.  2,040  sq.  rods. 

105.  How  many  square  acres  in  2,040  sq.  rods.  A.  12  A.  3  R. 

106.  Suppose  a  road  to  be  4  rods  wide,  how  many  acres  will  40 
rods  in  length  make  ?— will  1  mile  in  length  make '?— will  10  miles 
luake  ?  A.  I  acre ;  8  acres ;  80  acres. 

SOLID,    OR    CUBIC    MEASURE. 

107.  How  many  solid  inches  in  15  cords  of  wood  T 

108.  How  many  cords  of  wood  in  3,317,760  solid  inches "? 

109.  How  many  cubic  feet  of  earth  will  fill  a  cellar  15  feet  long, 
12  feet  wide,  and  8  feet  deep  1  (See  vii.   60.)  A.  1,440. 

110.  In  a  pile  of  wood  20  feet  long,  6  feet  high,  and  4  feet  wide, 
how  many  cord  feet  1 — ^how many  cords  ?         A.  30  C.  ft.;  3§  C. 

TIME. 

111.  How  many  seconds  are  there  in  15  years,  315  days,  23  hours 
and  57  seconds  -?  A.  500,338,857  seconds. 

112  How  many  more  seconds  in  a  leapyear  than  in  a  common 
year  of  365  days  ?  A.  86,400  seconds. 

113.  Suppose  your  age  to  be  15  years,  7  months,  3  weeks,  5  days, 
17  hours,  how  many  seconds  old  are  youl  A.  454,698,000  second. 

114.  How  many  seconds  has  it  been,  since  the  creation  of  the 
world,  to  the  close  of  the  year  A.  D.  1839,  allowing^the  birth  of 

Q.  How  are  furlongs  brought  into  degrees?  Miles  into  yards?  Barley  corns 
into  feet?  How  many  miles  in  5  leagues?  Inches  m  5  feet 2  mches?  What  is 
the  use  of  Square  Measure  ?  Repeat  the  Table.  How  are  the  square  contents 
obtained?  103.  How  many  square  yards  in  a  small  room  6  feet  square  JKoods 
in  a  piece  of  land  50  rods  long  and  2  rods  wide?  For  what  is  Solid  Measure 
used?  Repeat  the  Table.  How  are  cords  brought  into  solid  inches?  l<eet  into 
inches?  How  are  the  solid  contents  found?  109.  How  many  cord  feet  of  wood 
in  a  pile  8  feet  long,  4  feet  wide,  and  2  feet  high  ?  How  many  cord  feet  are  96 
solid  feet  of  wood?  For  what  is  the  Table  of  Time  used ?  Repeat  the  Table. 
How  may  centuries  be  reduced  to  days  ?  Days  to  years  ?  Seconds  to  hours  ( 
How  many  seconds  iu  120  minutes?  Minutes  in  1  h.  40  m.?— in  1^  h. . 


94  ARITHMETIC. 

Christ  to  have  taken  place  A.  M.  4,000,  and  each  year  to  contain 
365|days1*  A.   184,264,826,400  seconds. 

CIRCULAR    MOTION. 

115.  How  many  seconds  are  there  in  1  circle? 

116.  How  many  circles  in  1,296,000  seconds? 

117.  How  many  seconds  in  11  S.  15o  15^5^^? 

118.  How  many  signs  in  1,242,915  seconds'? 

TABLE    OF    PARTICULARS. 

1 19.  What  will  be  the  cost  of  420  dozen  eggs  at  1^  cents  for  each 
egg?  "  A.  $75.60. 

120.  When  buttons  are  5  mills  apiece,  what  will  50  dozen  cost  ? 
A.  $3.  What  will  50  gross  cost  ?  A.  $36.  What  will  50  great 
gross  cost?  A.  $432. 

121.  Suppose  2  hogs  to  weigh  40  score  and  15  pounds,  what  is  their 
value  at  5|- cents  per  pound  ?  A.  $44,825. 

122.  What  will  be  the  expense  of  200  reams  of  paper  at  25  cents 
per  quire?  A.  $1,000.     At  $.015  per  sheet  ?  A.  $1,440. 

123.  At  7  cents  a  pound,  how  many  barrels  of  beef  may  be  bought 
for  $15,050?  ^.  1,075  barrels. 

124.  At  5\  cents  a  pound,  what  will  1  quarter  of  flour  cost?  A. 
$1,31^.     What  will  196  pounds,  or  1  barrel  cost  ?  A.  $10.29. 


MISCELLANEOUS 

EXAMPLES   IN   REDUCTION. 

XLV.  1 .  In  80  guineas  how  many  dollars  at  6s.  each  ?  A.  $280. 

2.  In  224  boxes  of  sugar,  each  containing  27  lb.  how  many  hun- 
dred weight?  A.  60  cwt.  1  qr.  23  lb. 

3.  In  running  300  miles,  how  many  times  will  a  wheel  9  feet  2 
inches  in  circumference  turn  round  ?  A.   172,800  times. 

4.  In  172,800  turns  of  a  wheel  measuring  9  feet  2  inches  in  cir- 
cumference, how  many  miles  will  be  passed  over  ? 

5.  How  many  acres  on  the  surface  of  the  earth,  allowing  it  to 
contain  197,000,000  square  miles  ?        A.   126,080,000,000  acres. 

6.  How  many  times  does  a  clock  tick  in  a  leapyear,  supposing  it 
to  tick  once  every  second  ?  A.  31,622,400  times. 

7.  How  much  time  will  it  require,  for  a  man  that  is  worth  one 
million  of  dollars,  to  count  that  number,  at  the  rate  of  50  doUnrs  a 
minute,  supposing  him  to  be  employed  only  10  hours  each  day? 
A.  33  da.  3  h.  20  m. 

Q.  For  what  is  Circular  Motion  used?     Repeat  the  Table.     How  many  de- 
grees in  2  signs  ?     Minutes  in  2  degrees  ? — in  2f  j}  degrees  ? 

*  The  3  left  in  dividing  by  i  is  |  of  a  day,  or  because  24  hours  make  1  day,  it  is  |  of 
24  hours,  which  is  18  hours  to  be  added  in  when  multiplying  by  24  hours, 


\ 


MISCELLANEOUS    EXAMPLES.  95 

8.  Suppose  a  man  to  travel  39  miles  and  20  rods  a  day,  how  long 
would  it  take  him  to  travel  round  the  earth,  it  being  about  25,000 
miles  1  A.  1  Y.  275  days. 

9.  How  many  yards  of  carpeting  will  it  take  to  cover  the  floor  of 
two  parlors  each  18  feet  square  1  What  will  be  the  expense  at  $2\ 
per  yard  ^  * 

Note. — 18  feet  square  means  18  ft.  long,  and  18  feet  wide  ;  there- 
fore 18  ft.  X  18ft.  X  2  parlors^9  sq.  ft.  =72  sq.  yd.  A.  72  yd.;  $153. 

10.  Suppose  a  room  to  be  20  feet  square,  how  many  square  feet 
are  there  in  the  floor  ?  A.  400  sq.  ft. 

11.  How  many  square  feet  of  plastering  over  head,  in  a  room  20 
feet  square'?  A.  400  sq.  ft.  How  many  square  feet  in  one  side, 
supposing  the  room  to  be  12  feet  high  1  A.  240.  In  the  other  three 
sides?  A.  720  sq.  ft. 

12.  In  a  room  20  feet  long  and  16  feet  wide,  how  many  square 
yards  of  plastering  over  head  1  A.  35  sq.  yd.  5  sq.  ft.  How  many 
yards  of  carpeting,  1  yard  wide,  will  cover  the  floor  1 

A.  35  and  5  sq.  ft. 

13.  Suppose  the  foregoing  room  to  be  10  feet  high,  how  many 
yards  of  paper,  1  yard  wide,  will  coyer  one  end  1  A.  17  sq.  yd.  and  7 
sq.  ft.  How  many,  the  other  end?  A.  17  and  7sq.  ft.  How  many 
to  cover  both  sides  1  A.  4i  sq.  yd.  and  4  sq.  ft. 

14.  How  many  square  yards  are  there  in  the  floor  of  a  church, 
which  is  80  feet  long  and  67  feet  wide?       A.  595  sq.  yd.  5  sq.  ft. 

15.  How  many  shingles  18  inches  long,  4  inches  wide,  wiU  it  take 
to  cover  one  side  of  a  roof,  45  feet  long  and  25  feet  wide  ? 

Note. — In  laying  shingles,  two-thirds  of  the  length  of  each  shingle 
are  overlaid  by  others ;  therefore,  each  shingle  must  Be  reckoned  as 
covering  only  6  inches  in  length  and  four  inches  in  breadth,  making 
24  square  inches ;  then,  45  x  25  x  144  sq.  in.  ^24  sq.  in.  A.  6,750. 

16.  How  many  shingles  18  inches  long  and  4  inches  wide,  will  be 
required  to  cover  one  side  of  a  roof  60  feet  long  and  25  feet  wide  ? 
J..  9,000  shingles.     How  many  for  both  sides  1  A.  18,000  shingles. 

17.  On  a  certain  wharf  there  lies  a  pile  of  wood  40  rods  long,  6 
feet  high  and  4  feet  wide ;  how  many  cord  feet  will  it  make  ? 

A.  990  C.  ft. 

18.  How  many  times  wiU  a  wheel  which  is  15  feet  9  inches  in 
circumference,  turn  round  in  going  378  feet  1  A.  24.  In  going  from 
Providence  to  Norwich,  it  being  45  miles  ?  A.  15,085.+ 

19.  Suppose  a  farmer  rents  a  plantation  of  400  acres,  of  which  no 
more  than  200  are  tilled,  how  many  poles  are  there  in  the  remainder  1 

A.  32,000  poles. 

20.  In  a  lunar  month  of  27  days,  7  hours,  43  minutes,  5  seconds, 
how  many  seconds  1  A.  2,360,585  seconds. 

*  Note.— Observe  that  feet  multiplied  by  feet,  make  square  feet ;  inches  by  inches, 
square  inches,  &c.  Also,  that  square  inches  must  be  divided  by  square  inches ;  square 
feet  by  square  feet,  &c. 


96  ARITHMETIC. 

21.  How  many  seconds  is  it  from  the  birth  of  our  Saviour  to 
Christmas,  1828,  allowing  the  year  to  contain  365|  days,  or  365 
days  and  6  hours'?  A.  57,687,292,800  seconds. 

22.  When  a  person  is  21  years  old,  what  is  his  age  in  seconds  1 

A.  662,709,600  seconds. 

23.  The  wars  of  Bonaparte  caused,  as  is  computed,  in  20  years, 
the  deaths  of  at  least  2,103,840  persons  ;  how  many  would  that  be 
for  every  hour  of  the  20  years.        »  ^.  12  persons. 


3  . 

4  . 
6  9  . 

.       6 

0 

1  2 

.  5 

.  6 
.  0 

7  6  . 

1  8 

.11 

COMPOUND    ADDITION. 

XLVI.  1,  A  man  paid  10  shillings  for  a  gallon  of  oil,  15  shillings 
for  a  vest,  and  17  shillings  for  a  pair  of  boots  ;  how  many  pounds  did 
he  pay  for  the  whole?  A.  £2.  2s. 

2.  How  many  pounds  are  9s.,  16s.,  19s.,  and  lis.  *?  A.  jC2.15s. 

3.  In  one  lot  are  36  roods,  in  another  57  roods,  in  another  25 roods, 
and  in  the  fourth  17  roods  ;  how  many  acres  do  all  the  lots  contain? 

A.  33  A.  3  R. 

4.  Hence,  pounds  micst  be  added  to  pounds,  shillings  to  shillings, 
miles  to  miles,  ^c. 

5.  A  farmer  bought  a  load  of  hay  for  £Z.  6s.  5d.;  a  cow  for  £A.  6d.; 
and  a  horse  for  £69.  12s.;  what  did  he  pay  for  the  whole  ? 

Write  pence  under  pence,  shillings  under 
shillings,  &c.,  supplying  vacant  places  with 
ciphers,  then  add  up  each  column  as  in 
whole  numbers. 

6.  What  is  the  amount  of  i:i,583.  2s.  4d.  Iqr.;  i:2,036.  10s.  l|d., 
JC806.4S.  3d.;  ^£456.  Is.?  A.  £^,QSl.  17s.  8|d. 

7.  A  gentleman  purchased  four  loads  of  hay  weighing  as  follows^ 
viz.:  the  first  19cwt.  Iqr.  61b.;  the  second  17cwt.  lOlb,;  the  third 
18cwt.  2qr.  41b.;  and  the  fourth  220-^^.  31b.;  what  was  the  weight  ot 
the  whole?  A.  76cwt.  3qr.  231b. 

8.  What  is  the  sura  of  JETI 16.  12s.  3id.;  £13.  19s.  10id.;£4.  8^; 
£18.  4s.;  £905.  17s.  9d.;  £801. 14s.;  £9.  2d.  ? 

9.  The  column  of  farthings 
makes  6qr.  -4-4qr.  =  Id.  2qr ;  carry 
the  Id.  to  the  column  of  pence. 

The  pence  make  33d.^l2d.= 
2s.  9d. ;  carry  the  2s.  to  the  column 
of  shillings.  The  shillings  make 
68s. -^  20s.  =£3.  8s.;  carry  the  £3 
to  the  pounds,  which  add  as  in 
whole  numbers. 

XLVI.  Q.  In  adding  compound  numbers  how  do  you  proceed?  5.  What  is  to 
be  done  when  a  column  of  farthings  makes  6,  for  instance  ?  9. 


£. 

s. 

d. 

qr. 

116. 

1  2  . 

3 

.   1 

1  3  . 

1  9  . 

1  0 

.  2 

4  . 

0  . 

8 

.  3 

1  8  . 

4  . 

0 

.  0 

9  0  5. 

1  7  . 

9 

.  0 

8  0  1. 

1  4  . 

0 

.  0 

9  . 

0  . 

2 

.  0 

JL.  1  8  6  9  . 

8  . 

9 

.  2 

COMPOUND    ADDITION.  97 

10.  Hence,  divide  the  sum  of  each  column,  when  it  can  he  done,  as 
in  Reduction ;  write  down  the  remainder  and  carry  the  quotient  to 
the  next  column. 

11.  Find  the  sum  of  je205.  13s.  4|d.;  jGail.  15s.  8|d.;  £QQ.  10s. 
3^;  £49.  6|d.  A.  £535.  19s.  lOfd. 

12.  Find  the  sum  of  10  C.  27  Y.  8mo.  Iwk.  2d.  17h.  40m.  SOsec; 
85  C.  49  Y.  6mo.  6d.  15h.  50m.  SOsec;  65  C.  99  Y.  5mo.  5d.  lOh. 
27m.  45sec. 

The  first  column  makes 
125sec.-^60sec.=2ra.  5 
sec;  carry  the  2  minutes, 
&c. 


C.      Y.  mo. 

wk.  d. 

h.      m. 

sec. 

10.2  7.8. 

,1.2. 

1  7.4  0 

.3  0 

8  5.4  9.6. 

,0.6. 

15.50 

.  5  0 

6  5.9  9.5. 

,0.5. 

1  0.2  7 

.  4  5 

A.  16  1.76.7.3.0.19.59 


13.  When  the  sum  of  any  column  is  too  small  to  he  divided  as  ahove^ 
write  doivn  its  entire  sum  and  carry  none. 

14.  Find  the  sum  of  3171b.  lloz.  13d\vt.  15gr.;  2951b.  8oz.  2dwt. 
14gr.;  6151b.  8oz.  16gr.;  8191b.  8gr.  A.  2,0481b.  3oz.  17dwt.  5gr. 

15.  Add  together  £215.  8s.  2^;  £425.  6s.  8|d.;  £425.  id.;  £819 
2s.  5d.  A.  £1,884.  17s.  4id. 

RECAPITULATION. 

16.  Compound  Addition  is  the  adding  of  compound  numbers  ot 
the  same  kind  or  general  class. 

RULE. 

17.  Write  the  same  denominations  under  each  other. 

18.  Add  up  the  first  right-hand  column  and  divide  its  sum  hy  as 
many  of  that  denomination  as  make  1  of  the  next  greater  denomination. 

19.  Write  doiun  the  remainder  and  carry  the  quotient  to  the  next 
column,  proceeding  thus  to  the  last  column,  which  add  as  in  whole 
numbers. 

20.  The  PROOF  is  the  same  as  in  Simple  Addition. 

ENGLISH    MONEY. 

21.  Add  together  £17. 13s.  ll|d.;  £13. 10s.  2id.;  £10. 17s.  3|d.; 
£7.  7s.  Old.;  £2.  2s.  3id.;  £18.  17s.  lO^d.        A.  £70.  9s.  Ud. 

22.  Add  together  £8.  10s.  3id.;  £4.  9s.  8d.;  £1.  9s,  l^d.;  £2.  8s. 
7d.;  £4.  9s.  O^d.;  £8.  5s.  4|d.  A.  £29.  12s.  7d. 

TROY    WEIGHT. 

23.  Add  together  7501b.  9oz.  17dwt.  29gr.;  4501b.  6oz.  lldwt. 
llgr.;  8911b.  7dwt.  and  5391b.  3oz.  13dwt.  Igr. 

A.  2,6311b.  8oz.  9dwt.  17gr. 

24.  A  goldsmith  bought  four  ingots  of  silver,  the  first  of  which 
weighed  81b.  2oz.  12dwt,;  the  second,  5lb.  4oz.  5dwt.;  the  third,  6lb. 

Q.  What  is  the  general  direction  for  such  cases  ?  10.  When  the  sum  of  any 
column  is  less  in  value  than  one  of  the  next  higher  denomination,  how  do  you 
proceed?  13.  What  is  Compound  Addition  ?  16.  What  is  the  Rule  ?  17,  18,  19 
What  is  the  Proof?  20. 


9B 


ARITHMETIC. 


lOoz.  lldwt.;  and  the  fourth,  61b.  lloz.  15dwt.;  what  was  the  weight 
of  the  whole  1  A.  271b.  5oz.  3dwt. 

AVOIRDUPOIS    WEIGHT. 

25.  Add  together  2cwt.  3qr.  191b.  5oz.  7dr.;  Icwt.  2qr.  16lb.  4oz 
6dr.;  3cwt.  1  qr.  15lb.  2oz.  3dr.;  5cwt.  2qr.  121b.  loz.  6dr.;  2cwt. 
2qr.  141b.  4dr.;  5cwt.  Iqr.  151b.  2oz.  8dr. 

A.  21cwt.  2qr.  171b.  Idr. 

26.  A  grocer  sold  four  hogsheads  of  sugar,  weighing  as  follows  : 
the  first,  7cwt.  Iqr.  141b.;  the  second,  5cwt.  2qr.  lOlb.;  the  third, 
9cwt.  Iqr.  151b.;  the  fourth,  7cwt.  Iqr.  101b.;  what  did  the  whole 
weigh?  A.  29cvvt.  2qr.  241b. 

apothecaries'  weight. 

27.  Find  the  sum  of  171b.  5f.  25.  l9.  3gr.;  19lb.  2f.  75.  29. 
17gr.;  65ib.  llf.  45.  19gr.  and  75lb.  3f.  33.  13.  8gr. 

A.   1771b.  llf.  23.  7gr. 

28.  What  is  the  compound  formed  from  the  following  ingredients, 
viz.  :  5ib.  2f.  3^.  13.  12gr.  of  calomel;  3ib.  lOf.  53.  15gr.  of  jalap, 
71b.  8f.  73.  23.  14gr.  of  rhubarb,  and  lib.  3f.  23.  15gr.  of  the  ex- 
tract of  colocynth!  A.  18ib.  If.  23.  23.  16gr. 

cloth  measure. 

29.  Add  together  70yd.  2qr.  Ina.;  12yd.  Iqr.  Ina.;  9yd.  Ina.; 
40yd.  2qr.  Ina.;  56yd.  Iqr.  Ina.;  and  48yd.  Iqr.  Ina.  A.   237yd.  2na. 

30.  How  many  yards  are  565yd.  3qr.;  275yd.  3na.;  425yd.  Iqr. 
Ina.;  915yd.  2na.;  617yd.  2qr.  2na.,  and  719yd.  Iqr.  3na. 

A.   3,518yd.  Iqr.  3na. 

LONG    measure. 

31.  How  many  yards  are  617yd.  1ft.  lOin.;  810yd.  2ft.  llin.;  6yd. 
7in.;  85yd.  2ft.  5in.;  and  679yd.  3in.  1  A.  2,199yd.  2ft. 

32.  What  is  the  sum  of  the  following  distances ;  5401.  Im.  3fur. 
15rd.;  6401.  7fur.  39rd.;  7201.  2m.  3fur.  20rd.  7991.  39rd.;  5601.  Ifur. 
17rd.;  and  7501.  2m.  6fur.  23rd.  A.  4,0111.  1  m.7fur.  33rd. 

LAND    OR    SQUARE    MEASURE. 

33.  Add  together  45yd.  8ft.  113in. ;  45yd.  3ft.  112in. ;  75yd.  8ft. 
139in. ;  49yd.  115in. ;  and  589yd.  8ft.  90in.  A.  806yd.  3ft.  137in. 

34.  Find  how  many  acres  are  367A.  2R.  30rd. ;  815A.  IR.  16rd. ; 
50A.  2R.  20rd. ;  and  60A.  2R.  36rd.  A.  1,294A.  IR.  22rd. 

SOLID    OR    CUBIC    MEASURE. 

35.  Add  together  25T.  39ft.  1600in. ;  42T.  13ft.  1213in. ;  49T. 
25ft.  895in. ;  and  60T.  1689in.  of  round  timber. 

A.  177T.  30ft.  21  Sin. 

36.  Find  how  many  cords  are  189C.  127ft.  1500in. ;  3421C.  6ft. 
1720in.;  814C.  32ft.  815in. ;  617C.  96ft.  1629in. ;  915C.  915in. ; 
lOlC.  120ft.  1700in.;  83lC.  16ft.  250in. ;  and  901C.  113ft.  876in. 

A.  7,793C.  3ft.  764m. 


COMPOUND    SUBTRACTION.  99 

DRY    MEASURE. 

37.  How  many  bushels  are  715bu.  3pk.  7qt.  Ipt. ;  695bu.  Ipk. 
3qt.  ;  789bu.  2pk.  2qt.  ;   150bu.  3qt.  Ipt.  ;  167bu.  Iqt.  Ipt.  1 

A.  2,518bu.  Iqt.  Ipt. 

38.  Add  together  40bu.  2pk.  6qt.  Ipt. ;  89bu.  Ipk.  3qt. ;  75bu. 
2pk.  Iqt.  Ipt.  ;  69bu.  2pk.  3qt. ;  49bu.  Ipk.  2qt.  Ipt. ;  and  65bu 
3pk.  Iqt.  Ipt.  A.  390bu.  Ipk.  2qt. 

WINE    MEASURE. 

39.  Add  together  38gal.  2qt.  Ipt.  2gi. ;  16gal.  Iqt.  3gi. ;  20gal 
2qt.  Ipt.  Igi.  ;   18gal.  Iqt.  Ipt. ;  7gal.  Iqt.  2gi. ;  and  SOgal.  2qt.  Ipt. 

A.   132  gallons. 

40.  Find  the  sum  of  the  following  quantities:  615T.  Ip.  Ihhd. 
62gal.  2qt.  Ipt.  3gi.  ;  700T.  Ip.  Ihhd.  49gal.  Iqt.  Ipt.  Igi.  ;  513T. 
61gal.  2qt.  Ipt. ;  718T.  Ip.  Ihhd.  22  gal.  2qt.  Ipt.  3gi.;  and  871T 
38gal.  Iqt.  A.  3,420T.  45gal.  2qt.  Ipt.  3  gi. 

ALE    OR    BEER    MEASURE. 

41.  Add  together  15hhd.  42gal.  2qt.  Ipt. ;  75hhd.  39gal.  Iqt.  Ipt. ; 
62hhd.  15gal.  Iqt.  Ipt. ;  and  39hhd.  17gal.  Iqt.  Ipt. 

A.   193hhd.  6gal.  3qt. 

42.  How  many  barrels  are  17bl.  Ikil.  Ifir.  8gal.  2qt.  Ipt. ;  89bl. 
Ikil.  Ifir.  3gal.  Iqt.  Ipt.  ;  65bl.  Ifir.  6gal.  2qt.  Ipt.  ;  and  29bl.  Ikil. 
Ifir.  5gal.  3qt.  Ipt.  A.  203bl.  6gal.  2qt. 

TIME. 

43.  What  is  the  sum  of  5C.  64Y.  364d.  23h.  40m.  ISsec. ;  3C. 
19Y.  125d.  17h.  39m.  12sec. ;  4C.  85Y.  189d.  Uh.  13m.  23sec. ; 
7C.  45Y.  118d.  3h.  25m.  37sec. ;  9C.  63Y.  149d.  12h.  12m.  12sec. ; 
and  8C.  8h.  8sec.  A.  38C.  78Y.  218d.  4h.  10m.  47sec. 

44.  Find  the  amount  of  the  following  periods  of  time : — 49 Y. 
llmo.  3wk.  6d.  llh.  59m.  39sec. ;  45Y.  8mo.  2wk.  5d.  14m. 
42sec.;  65Y.  5mo.  Iwk.  3d.  19h.  25m.  Usee. ;  40Y.  3mo.  3wk.  2d. 
I7h.  11m.  4sec. ;  and  18Y.  Imo.  3wk.  Id.  21h.  8m.  8sec. 

A.  219Y.  7mo.  2wk.  5d.  21h.  58m.  44sec. 

CIRCULAR    MOTION. 

45.  Add  together  12S.  290.  59^.  59^^  45S.  15o.  45^  42^^;  65S. 
l8o.  IV.  W  and  62S.  13°.  19^  17^^.         A.'  186S.  17o.  16^.  38^^. 

46.  Find  the  sum  of  llS.  29o.  16^.  59^^;  20o.  4y.  IV';  8S.  3o. 
10^.  50^/  and  3S.  lOO.  6^  10^^.  A.  24S.  3o.  19^.  10^^ 


COMPOUND    SUBTRACTION. 

XLVH.  1.  Suppose  you  owe  £1  or  20  shillings,  and  pay  7  shil 
lings,  how  many  shillings  remain  unpaids  A.  13  shillings. 

2.  Suppose  you  draw  5  gallons  from  1  hogshead  of  molasses,  how 
touch  will  remain  in  the  hogshead  1  A.  58  gallons. 


100 


ARITHMETIC. 


3.  From  1  year  take  118  days.  A.  247  days. 

4.  Hence,  ive  must  subtract  shillings  from  shillings,  pence  from 
pence,  days  from  days,  dfc. 

£.       s.       d.     qr.         5.  From  £5.   13s.  6|d.  take  £3.  4s.  3|d. 
5  .   13.6.2      Begin  on  the  right  hand,  and  say,  Iqr.  from 

^ ^  '  ^  •   t      2qr.  leaves  Iqr.;  3d.  from  6d.  leaves  3d.  &c 

^  •       9.3.1  A.  £2.  9s.  3M. 


6.  A  horse  that  cost  jCl7.  13s.  9fd.,  was  sold  for  only  £10.  3s. 
42d. ;  what  was  the  loss  upon  it  1  A.  £7.  10s.  5^. 

7.  From  8037hhd.  40gal.  2qt   Ipt.  3gi.,  take  7948hhd.  29gal.  2qt. 
Ipt.  1  gi.  A.  89hhd.  llgal.  2gi. 

8  From  £:8.  15s.  2d.  take  £3.  9s.  8d. 

£.       s.       d.  Borrow  ls.  =  12d.,  which  added  to  2d.  makes 

8  •  1  5  •  2  i4d. ;  then  say,  8d.  from  14d.  leaves  6d. ;  carry- 
^  •  9  '  8  jjjg  Is  (borrowed)  to  9s.  =  10s.  from  15s. =5s.; 
5  .       5.6      £3  from  £8  leaves  £5. 


9.  Hence,  we  may  borrow  one  of  the  next  higher  denomination  and 
add  its  value  in  the  next  lower  denomination  to  the  upper  figure,  then 
subtract  as  before  and  carry  the  1  to  the  lower  figure  in  the  next 
column. 

10.  From  £^18.  17s.  6d.  take  i^ll.  9s.  8d.         A.  £7.  7s.  lOd. 

11.  From  43  hours  23m.  30sec.  take  19h.  5m.  40  sec. 

Note. — Add  60  sec.  to  the  30  sec,  or  we  may  subtract  the  40  sec. 
from  the  60  sec.  first,  and  add  the  30  sec.  to  the  remainder ;  thus, 
40  sec.  from  60  sec.  =20  sec. +  30=50  sec.     A.  24h.  17m.  50sec. 

12.  From  813C.  102ft.  lOOOin.,  take  787C.  35ft.  1727in. 

A.  26C.  66ft.  1001  in. 

13.  From  812hhd.  23gal.  Iqt.  take  716hhd.  29  gal.  2  qt. 

hhd.        gal.    qt.          Say,  2qt.  from  4qt.=2qt.  + Iqt.  =3qt.     Next, 
8  12.23.1      1  (to  carry)  to  29 =30gal.  from  63  gal.  =33 +23 
7  16.29.2     =56  gal. ;  then  carry  1  to  the  716. 
9  5.56.3 


14.  From  514hhd.  63gal.  2qt.,take  235hhd.  55gal.  3qt. 

A.  278hhd.  60gal.  3qt. 

15.  From  817m.  4fur.  22rd.,  take  619m.  6fur.  17rd. 

A.  197m.  6fur.  5rd. 

16.  From  1  tun  take  46  gallons  and  2  quarts. 

T.     p.  hhd.    gal.     qt.         17.  Say,  2  from 4  (qt.)  leaves  2.   1  (to 
1.0.0.       0.0      carry)  to  46  makes  47,  which  from  63 

"^6.2      (gal.)  leaves   16 ;    1   (to  carry  again) 

A.  1.1.16.2      from  2  (hhd.)  leaves  1  hhd.  &c. 

XLVII.  Q.  How  do  you  subtract  numbers  of  different  denominations  ?  4. 
In  subtracting  for  instance,  8d.  from  15s.  2d.  &c.  how  do  you  proceed?  8. 
How  are  40  seconds  subtracted  from  30  seconds  23  minutes,  &c.?  11.  Wha* 
other  method  produces  a  like  result?  11. 


COMPOUND    SUBTRACTION.  101 

18.  From  785hhd.  take  696hhd.  29gal.  2qt.  Ipt.  3gi. 

A.  88hhd.  33gal.  Iqt.  Igi. 

19.  From  87,563  yards  take  1  nail.         A.  87,5G2yd.  3qr.  3na. 

-     RECAPITULATION. 

20.  Compound  Subtraction  is  the  subtracting  of  one  compound 
number  from  another  of  the  same  kind  or  general  class. 

RULE. 

21.  Write  the  smaller  quantity  under  the  greater,  with  the  same 
denominations  under  each  other ;  then  begin  on  the  right  and  sub- 
tract the  numbers  in  each  denomination  separately  as  in  Simple  Suh- 
tr  action. 

22.  But  when  a  lower  number  in  any  denomination  exceeds  the  one 
over  it,  add  to  the  upper  number  as  many  units  as  make  one  of  the 
next  higher  denomination,  from  which  subtract  as  before,  and  carry  I 
to  the  next  lower  number. 

23.  The  PROOF  is  the  same  as  in  Simple  Subtraction. 

ENGLISH     MONEY. 

34.  From  ^3..5s.  4d.  take  £\.  2s.  8d.  A.  £2.  2s.  8d. 

25.  Suppose  a  gentleman  has  JGIOO,  and  gives  jC19.  5s.  4^.  for 
his  passage  to  England ;  how  much  will  he  have  left  on  his  arrival 
there  ?  A.  JCSO.  14s.  7^d. 

TROY    WEIGHT. 

26.  From  31b.  5oz.  lOdwt.  take  lib.  6oz.  13dwt. 

A.  lib.  lOoz.  17dwt» 

27.  A  gentleman  has  a  silver  teapot  weighing  31b.  7oz.  5dwt. 
22gr.,  and  a  silver  cup  weighing  21b.  lOoz.  13dwt.  15gr. ;  what  is 
the  difference  in  their  weight]  A.  8oz.  12dwt.  7gr. 

AVOIRDUPOIS    WEIGHT^ 

28.  From  lOT.  15cwt.  Iqr.  101b.  take  5T.  17cwt.  2qr.  221b. 

A.  4T.  17cwt.  2qr.  131b. 

29.  A  merchant  bought  two  hogsheads  of  sugar,  which  together 
weighed  16cwt.  3qr.  171b.  8oz.,  and  the  smaller  hogshead  weighed 
7cwt.  Iqr.  201b.  lOoz. ;  what  was  the  weight  of  the  larger  one  1 

A.  9cwt.  Iqr.  211b.  14oz. 

APOTHECARIES*    WEIGHT. 

30.  From  49Jb.  3?.  63.  l3.  take  llf.  63.  23.  5gr. 

A.  48ib.  3f.  63.  13.  I5gr. 

31.  Suppose  an  expectorant^  to  consist  of  2lb.  3|.  73.  13.  15gr. 
of  the  mucilage'  of  gum  arable,  and  lib.  9f.  43.  23.  19gr.  of  the 
oxymel  of  squill ;  how  much  is  there  of  one  quantity  more  than  of 
the  other"?  A.  6f.  23.  13.  16gr. 

Q.  What  is  Compound  Subtraction?  20.  What  the  Rule?  21,  22.   Proof?  23. 

1  Expectorant.    That  which  promotes  discharges  from  the  lungs. 
3  Mucilage.  A  slimy  or  viscous  body 
9* 


168  ARtTMMETlC. 

CLOTH    MEASURE. 

32.  From  810  English  ells  take  Iqr.  Ina.    A.  809E.e.  3qr.  3na. 

33.  A  merchant  bought  500yd.  2na.  of  broadcloth,  and  sold  412yd 
2qr. ;  how  much  had  he  left  1  A.  87yd.  2qr.  2na. 

LONG    MEASURE. 

34.  From  19yd.  1ft.  7in.  Ib.c.  take  6yd.  2ft.  2b.c. 

A.  12yd.  2ft.  6in.  2b.c. 

35.  Suppose  a  footman  goes  3m.  4fur.  17rd.  an  hour,  and  a  rail- 
road car  39m.  2fur.  20rd.  in  the  same  time ;  how  much  does  one 
gain  of  the  other  in  one  hour !  A.  35m.  6fur.  3rd 

LAND    OR    SQUARE    MEASURE. 

36.  From  657yd.  3ft.  lin.  take  398yd.  6in.  A.  259yd.  2ft.  139in. 

37.  If  from  a  field  containing  40A.  2R.  20rd.  there  be  taken  19A. 
3R.  30rd.,  how  much  will  there  be  left  1  A.  20A.  2R.  30rd. 

SOLID    OR    CUBIC    MEASURE. 

38.  From  17  tons  of  round  timber  take  1720  inches. 

A.  16T.  49ft.  8in. 

39.  Suppose  315C.  68ft.  of  wood  be  taken  from  a  pile  containing 
1000  cords ;  how  many  cords  will  be  left  1  A.  684C.  60ft. 

DRY    MEASURE. 

40.  Subtract  7bu.  2pk.  6qt.  from  12bu.  A.  4bu.  Ipk.  2qt. 

41.  A  farmer  having  raised  40  bushels  of  corn,  kept  23bu.  2pki 
for  his  own  use,  and  sold  the  remainder ;  what  quantity  did  he  sell  1 

A.  16  bushels  2  pecks. 

WINE    MEASURE* 

43.  From  3hhd.  15gal.  take  19gal.  3qt.       A.  2hhd.  58gal.  Iqt. 

43.  A  grocer  bought  5  hogsheads  of  molasses,  and  sold  Ihhd 
25  gals.;  how  much  had  he  then  on  hand?  A.  3hhd.  38gal. 

ALE    OR    BEER    MEASURE. 

44.  From  7bl.  Ifir.  take  Ikil.  3qt.  A.  6bl.  Ikil.  8gal.  Iqt. 

45.  Suppose  a  brewer  has  in  one  cellar  39bl.  Ikil.  Ifir.  5gal.  Iqt* 
of  beer,  and  in  another  25bl.  Ifir.  6gal.;  how  much  more  has  he  in 
one  cellar  than  in  the  other  1  A.  14bl.  Ifir.  8gal.  Iqt. 

TIME. 

46.  From  ly.  3mo.  2wk.  take  8mo.  3wk.  A.  6mo.  3wk. 

47.  Suppose  a  father's  age  is  45Y.  6mo.  3wk.  5d*.,  and  his  son's 
22 Y.  9mo.  Iwk.  6d.;  how  much  does  the  father's  age  exceed  the 
son's  1  A.  22Y.  9mo.  Iwk.  6d. 

CiRCttAR  MOTION. 

48.  From  298.  8o  take  21°  15^  30^^       A.  28S.  16o.  44^  30^^ 

49.  The  Moon  is  5S.  18°  14'  17/'  east  of  the  Sun,  and  Jupiter 
12S.  280  43/  45// ;  how  far  are  the  Moon  and  Jupiter  apart  1 

A.  7S.  100.  29' 28''. 


£. 
19  7  6. 

s. 
5  , 

d. 
.  3 

qr. 
.   1 
3 

6  9  2  8. 

1  5 

.  9 

.  3 

COMPOUND   MULTIPLICATION.  103 


COMPOUND  MULTIPLICATION. 

XLVIII.  I.  At  8  shillings  a  quarter,  how  many  pounds  will  pur- 
chase 5  quarters  of  flour?  A.  £2.  (=40s.) 

2.  Suppose  it  takes  3qr.  of  a  yard  of  cloth  for  one  vest,  how  many 
yards  will  be  required  for  12  vests  T  A.  9  yards. 

3.  How  many  bushels  are  8  times  3  pecks  ?  A.  6  bushels. 

4.  Suppose  a  bottle  to  contain  3  quarts  of  molasses,  how  many  gal- 
lons would  9  such  bottles  hold  ■?  A.  6gal.  3qt. 

6.  A  ship  is  valued  at  i?  1,976.  5s.  3|d.,  and  her  cargo  of  specie  at 
3  times  as  much;  how  much  specie  has  she  on  board  1 

Say,  3  times  1  qr.=3qr.,  3  times 
3d.=9d.,  &c. 

A.  jE:5,928.  15s.  9fd. 

6.  Hence  we  may  multiply  each 
denomination  separately,  as  in  simple 
numbers. 

7.  Multiply  £17,865.  3s.  Id.  by  4.  A.  jC31,460.  12s.  4d. 

8.  Multiply  346m.  Ifur.  6rd.  by  6.  A.  2,076m.  6fur.  36rd. 

9.  A  merchant  bought  5  yards  of  cloth  for  £2.  6s.  Ifd.;  what  was 
the  cost  of  the  whole  ? 

Say,  6  times  3qr.  =  15qr.-J-4qr.=3d. 
3qr.,  carry  the  3d.:  5  times  ld.=5d.+ 
3d.  =8d. :  5  times  6s.  =30s. -^20s.  =£1. 
lOs.  carry  the  £l. :  and  so  on. 

A.  £11.  10s.  8fd. 

10.  Hence,  carry  after  multiplying  as  in  Compound  Addition. 

11.  Multiply  £5.  8s.  Ud.  by  6.  A.  £^2.  8s.  7|d. 

12.  Multiply  £8.  10s.  6fd.  by  8.  A.     jC68.  4s.  6d. 

13.  What  is  the  product  of  105T.  Ip.  Ihhd.  37gal.  3qt.  Ipt.  3gi. 
multiplied  by  9  ? 

Say,  9x3gi.=27gi.-4- 

T.        p.  hhd.  gal.     qt.  pt.    gi.        4gi.=6pt.  3gi.;  carry  the 

105.1.1.37.3.1.3         6pt. :  9  times  lpt.=9pt. 

^         +6pt.  =  15pt.^2=7qt., 

4-  9  53.0.0.26.2.1.3         ipt.;  carry  the  7qt.  and 

so  on. 

14.  Multiply  201.  2m.  5fur.  15rd.  by  10.  A.  2081.  2m.  5fur.  30rd. 

15.  Multiply  151b.  5oz.  13dwt.  by  11.  A.  1701b.  2oz.  3dwt. 

16.  Multiply  I7bu.  Ipk.  3qt.  by  12.  A.  208bu.  Opk.  4qt. 

17.  When  the  multiplier  is  a  composite  number,  multiply  su4;ceS' 
lively  hy  its  factors.  See  Xvii.  8. 

18.  What  is  the  product  of  £2.  10s.  4|d.  multiplied  by  24?  3X8 
^24:  then  multiply  first  by  3  and  that  product  by  8.   A.  £60. 9s. 

XLVIII.  Q.  When  oats  are  2s.  6d.  per  bushel,  what  will  be  the  cost  in  shil 
kngs  and  pence  of  3  bushels?— of  4  bushels  ?--of  6  bushels  ? 


£. 

s. 

d. 

qr. 

2  , 

6  . 

,   1 

.  3 
5 

1  1  . 

1  0  . 

,  8  , 

.  3 

104  ARltHMETiC* 

19.  Multiply  5h.  10m.  SOsec  by  48.  A.  248h.  l^iii. 

20.  Multiply  5Y.  3mo.  3wk.  by  96.  A.  510  years. 

21.  When  the  multiplier  is  not  a  composite  number,  multiply  by  the 
whole  of  it  at  once. 

£  .      s.  d.  ^^-  What  will  95  pair  of  slippers  cost 

0  .       2  .       6         at  2s.  6d.  a  pair? 

9_5  Say,  95  times  6d.  =  570d.-^12d.=47s 

A.  £l   1  .  1  7  .       6         6d. :  95x2s.  =  190s.+47s.=237s.-e-20=i 
======        £11.  17s. 

23.  At  7s.  4^d.  per  bushel,  what  will  23  bushels  of  wheat  cost  ? 

A.  £8. 9s.  7^. 

24.  Suppose  it  takes  2gal.  Iqt.  Ipt.  2gi.  to  fill  a  demijohn,  how 
much  would  be  required  to  fill  19  such  vessels  T 

A.  46gal.  Iqt.  Opt.  2gi. 

RECAPITULATION. 

25.  Compound  Multiplication  is  the  multiplying  of  a  compound 
number  by  a.simple  one. 

RULE. 

26.  Multiply  each  denomination  separately,  carrying  as  in  Cdm* 
pound  Addition. 

27.  Multiply  je819.  3s.  O^d.  by  8.  A.  :£:6,553.  8s.  2d. 

28.  What  will  be  the  cost  of  73  pair  of  shoes  at  5s.  6d.  per  pair  t 

A.  £20.  Is.  6d. 

29.  Multiply  51b.  lOoz.  17dwt.  by  9.  A.  531b.  loz.  13dwt. 

30.  How  many  pounds  will  7  cups  weigh,  when  one  weighs  31b. 
5oz.  13dwt.  llgr."?  A.  241b.  3oz.  14dwt.  5gr. 

31.  Multiply  3T.  15cwt.  Iqr.  151b.  by  13. 

A.  49T.  Ocwt.  Oqr.  20lb. 

32.  What  is  the  whole  weight  of  17  hogsheads  of  sugar  each  of 
Ivhich  weighs  12cwt.  Iqr.  201b.'?  A.  211cwt.  2qr.  151b. 

33.  Multiply  5ib.  3f.  65.  l3.  by  27.  A.  143lb.  6f.  35. 

34.  Suppose  a  box  of  pills  to  contain  a  compound  of  aloes  and 
jalap,  weighing  If.  33.  13.;  what  quantity  would  be  required  to  fill  1 
dozen  boxes  1  1  gross  of  boxes  ]  1  great  gross  of  boxes  ? 

A.  Dozen,  lib.  5|.;  gross,  171b.;  great  gross,  204ib. 

35.  Multiply  5yd.  3qr.  Ina.  by  8.  A.  46yd.  2qr. 

36.  Bought  21  pieces  of  broadcloth,  each  containing  13yd.  2qr. 
3na.;  how  many  yards  were  there  in  the  whole  ? 

A.  287yd.  Iqr.  3na. 

37.  Multiply  5m.  2fur.  7rd.  by  8.  A.  42m.  Ifur.  16rd. 

38.  Suppose  a  man  travels  20m.  5fur.  20rd.  in  one  day;  how  far 
would  he  travel  in  a  year  at  that  rate  1         A.  7,550m.  7fur.  20rd. 

39.  Multiply  8A.  3R.  lOrd.  by  108.  A.  951A.  3R. 

40.  Suppose  the  floor  of  a  spacious  hall  to  contain  200  sq.  yd.  5ft 

Q.  What  is  Compound  Multiplication?  25.  What  is  the  Rule  1  26. 


COMPOUND    DIVISION.  105 

75in.;  how  many  square  yards  would  it  contain,  if  it  were  13  times 
as  large?  A.  2,607yd.  8ft.  lllin. 

41.  Multiply  5T.  l,429in.  of  hewn  timber  by  144. 

A.  722T.  39ft.  144in. 

42.  There  are  24  piles  of  wood,  each  containing  3  cords  and  42 
cubic  feet;  what  quantity  do  all  the  piles  contain  ?  A.  79C.  112ft. 

43.  Multiply  819bu.  3gi.  by  11.    A.  9,009bu.  Opk.  4qt.  Opt.  Igi. 

44.  How  many  bushels  of  oats  are  there  in  6  bins,  in  each  of 
which  are  15  bags,  each  containing  3bu.  Ipk.  6qt.  Ipt.  1 

A.  SlObu.  3pk.  Iqt. 

45.  Multiply  9bl.  Ikil.  Ifir.  Bgal.  by  7.  A.  69bl.  Ikil.  Ifir.  2gal. 

46.  How  many  gallons  of  "hard  cider"  can  be  put  into  1,728  bot- 
tles, supposing  each  bottle  to  hold  3qt.  Ipt.  1        A.  1,512  gallons. 

47.  Multiply  7T.  Ip.  Ihhd.  20gal.  by  6. 

A.  46T.  Ip.  Ihhd.  57gal. 
"  48.  How  many  hogsheads  of  water  will  be  sufficient  to  supply  an 
army  of  50,000  men  for  one  day,  supposing  each  man  to  require  Iqt. 
Ipt.  Igi.!  A.  322hhd.  26gal.  2qt. 

49.  Multiply  5C.  89Y.  115d.  by  8.  A.  47C.  14Y.  190d. 

50.  The  sun  performs  his  rotation  on  his  axis  in  25d.  14h.  8m.; 
how  many  years  would  he  be  in  performing  450  such  revolutions  ? 

A.  31Y.  200d. 

51.  Multiply  5S.  29°.  4^  by  96.  A.  573S.  Qo  24^. 

52.  Suppose  one  ship  is  in  5°  15^  45^^  south  latitude,  and  another 
4  times  farther  south ;  what  must  be  the  latitude  of  the  latter  1 

A.  2  lo.  3^  south  latitude. 


COMPOUND  DIVISION. 

XLIX.  1.  When  5  bushels  of  wheat  cost  £2.  how  many  shillings 
will  purchase  1  bushell  (.£2= 40s.)  A.  8  shillings. 

2.  Suppose  8  boys  to  have  gathered  4  bushels  of  chestnuts,  how 
many  pecks  wUl  each  have  if  they  are  divided  equally  1  A.  2  pecks. 

3.  Suppose  a  ship's  cargo,  valued  at  £232.  16s.  8d.  to  be  owned 
equally  by  4  men ;  what  is  each  one's  part  1 
£.  s.       d. 

^  )  2  3  2  .  1  6  .  8  Divide  the  dC232  first  by  4 ;  then 

J..       £58.       4.2        the  16s.  by  4 ;  lastly  the  8d.  by  4. 

4.  Hence,  we  may  divide  each  denomination  separately,  as  in  simple 
numbers. 

5.  Divide  128161b.  12oz.  6dr.  by  6.  A.  2,136lb.  2oz.  Idr. 

6.  Divide  7  pounds  of  bread  equally  among  8  soldiers.    (Sep  Troy 
Weight.)  A.  lOoz.;  and  4oz.  left  (undivided). 

XLIX.     Q.  How  are  compound  numbers  divided  ?  4. 


106  ARITHMETid 

7.  Suppose  the  8  soldiers  wish  to  divide  the  4  ounces  also ;  how 
many  pennyweights  would  there  be  apiece  ?  A.  lOdwt. 

8.  Hence,  when  there  is  a  remainder,  we  may  reduce  it  hy  Reduc- 
tion Descending,  and  divide  again,  and  so  on. 

9.  Divide  £l\S  equally  among  6  persons. 

£  s.     d.  10.  Say,  £118-4-6=19  times  and  £4: 

^^  ^  ^  Q  •       Q  •  Q        left:  je4x20s.=80s.-^6  =  13   times  and 
A.     19.13.4        2s.  left:  2s.x  12d.=24d.-^6=4  times. 

11.  Divide  43cwt.  into  5  equal  parts.  A.  8cwt.  2qr.  101b. 

12.  Suppose  203  bushels  of  corn  will  fill  5  bins  of  equal  size,  what 
quantity  can  be  put  into  each  bin  ?  A.  40bu.  2pk.  3|qt. 

13.  A  father  divided  1300  acres  of  land  equally  among  his  7  sons ; 
what  quantity  did  he  give  to  each  ?  A.  185A.  2R.  34frd. 

14.  Divide  £2^.  14s.  by  9.  £.        s. 

.      15.  Thei:21eftx20s.=40s.tl4s.=54s.  9)29.14 

-i- 9 =6  times.  A.  3.6 


16.  Recollect  then,  when  a  remainder  is  brought  into  the  next  de- 
nomination, to  add  in  the  given  number  of  that  denomination. 

17.  Divide  £^1.  15s.  by  3.  A.  £\5.  18s.  4d. 

18.  Divide  67cwt.  Iqr.  101b.  into  5  equal  quantities. 

cwt.    qr.      lb.  The  2cwt.  over  X4qr.  =  8qr.  +  lqr=9qr. 

5)67.1.10       ^5  =  1  time  and  the  4qr.  over X 251b.  = 
A.         13.1.22       1001b. +  10=1101b.^5=22  times. 

19.  Divide  134cwt.  3qr.  19lb.  by  6.  A.  22cwt.  Iqr.  241b. 

20.  Divide  500Y.  3mo.  18d.  by  9.  A.  55Y.  7mo.  2d. 

21.  When  the  divisor  exceeds  12,  and  is  a  composite  number,  divide 
successively  by  its  several  factors.     See  xxiv.   1. 

22.  Divide  £84.  10s.  6d.  by  24.  A.  £3.  10s.  5ld. 

23.  Divide  155yd.  Iqr.  Ina.  by  35.  A.  4yd.  Iqr.  3na. 

24.  When  the  divisor  is  not  a  composite  number,  we  may  divide  by 
the  whole  divisor  at  once,  after  the  manner  of  Long  Division. 

25.  Divide  671  hogsheads  9  hhd.      gal. 

gallons  by  29.  2  9)671.9(23 


5  8 


Dividing  the  671hhd.  by  29,  as 
above  directed,  leaves  4hhd.,  which 
we  multiply  by  63  gallons,  and  add 
in  the  9  gallons,  making  261  gal- 
lons, to  be  divided  by  29,  as  at  first. 
A.  23hhd.  9gal. 


9  1 

8  7 

4 
6  3 

2  5  2 
9 

2  9) 

2  6  1 
2  6  1 

(9 

Q.  When  there  is  a  remainder,  how  do  you  proceed  ?  8.  Suppose,  for  in 
stance,  in  dividing,  there  i3  a  remainder  of  £4,  how  do  you  proceed  with  it?  10 
How  do  you  proceed  with  a  remainder  of  2  shillings  ?  10. 


COMPOUND    DIVISION. 


107 


26.  Divide  £332.  19s.  9d.  by  34.  A.  £9.  15s.  10|. 

27.  Divide  120A.  2R.  37rd.  by  47.  A.  2A.  2R.  llrd. 
28    Divide  483Y.  ISOd.  15h.  30m.  45sec.  into  105  equal  periods 

of  time.  A.  4Y.  227d.  16h.  8m.  51yV3sec. 

RECAPITULATION. 

29.  Compound  Division  is  the  dividing  of  a  compound  number  by 
a  simple  one. 

^  RULE. 

30.  Begin  on  the  left  and  divide  each  denomination  separately^  as 
in  simple  numbers. 

31.  But  if  there  be  a  remainder,  reduce  it  to  the  next  denomina' 
tion,  to  which  add  the  given  number  in  that  denomination,  then  divide 

as  before.  .  t  •    'j-  -j     j         j 

32.  Each  quotient  will  be  of  the  same  name  with  its  dividend ;  and 
the  several  quotients  taken  together  will  constitute  the  required  quo- 
tient or  answer.  I 

33.  Divide  i:i61.  14s.  4d.  by  8.  A.  £20.  4s.  3^. 

34.  If  a  man  can  earn  £3.  14s.  5id.  ^er  week,  how  much  can  he 

earn  per  day]  .\^^!'^^?Pl' 

35.  Divide  301b.  7oz.  13dwt.  by  9.  A.  31b.  4oz.  17dwt. 

36.  When  7  silver  cups  weigh  81b.  9oz.,  what  is  the  weight  ot 
each?  Allb.3oz. 

37.  Divide  205qr.  19lb.  7oz.  2dr.  by  10. 

A.  20qr.  141b.  7oz.  l^dr. 

38.  Suppose  a  poor  man  labors  a  month  for  1491b.  13oz.  of  pork ; 
how  much  does  he  receive  each  day,  on  an  average,  allowing  26  work- 
ing days  to  each  month  T  ^;  51b.  12^j0z. 

39.  Divide  853  yd.  2qr.  3na.  by  157.  A.  5yd.   Iqr.  3na. 

40.  If  it  take  2,700  yards  of  broadcloth  to  clothe  a  regiment  of  800 
men,  what  quantity  will  each  man  require  1  A.  3yd.  Iqr.  2na. 

41.  If  a  hogshead  of  wine  costs  £45.  8s.  3d.,  what  is  it  worth  by 

th        Uonl  ^"  ^^®' 

42^  Bought  2  dozen  (24)  silver  spoons,  which  weighed  71b.  6oz. 
13dwt.-  how  much  silver  did  each  spoon  contain  \  ,        ,  „ 

A.  3oz.  15dwt.  13gr. 

43.  Suppose  a  steamboat,  in  making  121  trips  from  Albany  to  New 
York,  occupies  48d.  17h.  40m.;  what  will  be  the  average  time  m 
which  she  makes  one  trip  1  ^,  f'  9^-  ^Om- 

44  How  far  must  I  travel  each  day,  to  accomplish  a  journey  ot 
1,400  miles  3fur.  lOrd.  in  51  days]  ^".^^i^i  ^^^'-  f  ?'^  .r 

45.  Suppose  37  barrels  of  equal  size  contain  98bu.  3pk.  ^qt.  ot 
wheat ;  what  quantity  is  in  each  barrel  ]  A.  2bu  2pk.  bj^gt. 

Q.  When  there  is  a  remainder,  what  is  to  be  done  with  each  i^^f  i^^-J^^JJ^^- 
naS^on  of  the  dividend  ?  16.  How  do  you  proceed  when  the  divisor  exceeds  12, 
and  is  a  composite  number?  21.  How,  when  it  is  not  a  composite  number?  24 
What  is  Compound  Division?  29.  Rule?  30,  31,  32, 


108  ARITHMETIC. 

46.  Suppose  a  king's  salary  to  be  £200,000  per  annum ;  what  is 
that  per  day?  A.  £547. 18s.  lOd.  3^|qr. 


MISCELLANEOUS   EXAMPLES. 

L.  1.  How  many  farthings  are  there  in  £5. 17s.  6d.  ? 

2.  How  many  pounds  are  there  in  5,640  farthings  I 

3.  How  many  guineas  at  28  shillings  each,  will  pay  a  debt  of  £25 1 

A.  17  guineas  24  shillings, 

4.  A  grocer  bought  20  hundred  weight  of  sugar  for  $112,  and  sold 
it  for  4^d.  per  pound  ;  what  was  the  gain  1  ^4.  $13. 

5.  A  merchant  in  London  borrowed  £60  and  paid  at  one  time  jCl5. 
14s.  6d.,  and  at  another  jG20.  3s.  6|d.,  how  much  remained  unpaid  1 

A.  £24.  Is.  ll|d. 

6.  From  a  compound  weighing  5lb,  an  apothecary  sold  to  one  man 
lib,  3f.  55.  13.,  and  to  another  3f.  23.,  how  much  had  he  left  on 
hand?  A.  31b.  5f.  25. 

7.  A  merchant  bought  3  hogsheads  of  sugar,  each  weighing  8cwt. 
Iqr.  201b.,  and  sold  five  barrels  of  the  same,  each  weighing  3cwt.3qr 
171b.     How  much  had  he  left?  A.  5cwt.  3qr. 

8.  If  you  deduct  the  days  in  the  months  of  November  and  Decem- 
ber from  the  year,  how  many  days  will  there  be  left  in  a  leapyear  ? 

A.  305  days. 

9.  What  is  the  sum  of  30rd.  5yd.,  19rd.  4yd.,  17rd.  1yd.,  and  25rd. 
4yd.  ?    See  xli.  16,  17.  A.  93rd.  3yd. 

10.  Add  together  30A.  2R.  39  sq.  r.  30  sq.  yd.,  29A.  IR.  25  sq.  r 
23  sq.  yd.,  16A.  3R.  8  sq.  r.  15  sq.  yd.,  and  45A.  27  sq.  r.  8  sq.  yd. 

A.   122A.  21  sq.  r.  15f  sq.  yd. 

11.  Add  into  one  sum  500  sq.  r.  272  sq.  ft.,  450  sq.  r.  195  sq.  ft., 
365  sq.  rd.  215  sq.  ft.,  and  985  sq.  r.  270  sq.  ft. 

A.  2,303  sq.  r.  135|  sq.  ft. 

12.  If  a  man  travels  25m.  3fur.  15rd.  3yd.  a  day,  for  12  successive 
days,  how  far  will  he  go  in  that  time  ?  A.  305m.  26rd.  3yd. 

13.  From40rd.  2yd.  take  17rd.  4yd.  Say  4yd.  from  5iyd.  =  l^yd. 
+2yd.=3i  and  carry  1.  A.  22"rd.  3^yd. 

14.  Add  together  22rd.  3jyd.,  and  17rd.  4yd.        A.  40rd.  2yd. 

15.  Suppose  a  man  travels  305m.  26rd.  3yd.  in  12  days  ;  what  is 
the  average  distance  per  day?  A.  25m.  3fur.  15rd.  3yd. 

16.  How  many  gallons  in  50bl.  25gal.  A.  1,600 

17.  How  many  barrels  in  1,600  gallons  ?  A.  50bl.  25gal. 

18.  How  many  pint,  quart  and  2  quart  bottles,  of  each  an  equal 
number,  can  be  filled  with  a  hogshead  of  molasses  ? 

Note. — 4pt.  [=2qt. :]  2pt.  [=lqt.]  and  Ipt.  make  7  pints;  then 
divide  63  gallons  brought  into  pints  by  7  pints.         A.  72  of  each. 

19.  A  merchant  has  700  quart,  700  two  quart,  700  three  quart  and 


FRACTIONS.  109 

700  gallon  bottles,  and  wishes  to  know  how  many  hogsheads  of  wine 
it  will  take  to  fill  themT  A.  27hhd.  and  49gal.  over. 

20.  A  certain  manufacturer  employs  an  equal  number  of  men,  boys 
and  girls,  to  whom  he  pays  daily,  as  follows,  viz  :  to  each  man  $1,  to 
each  boy  50  cents,  and  to  each  girl  75  cents,  making  in  all  $6  75. 
How  many  persons  of  each  class  has  he  in  his  employ  1 

A.  300  persons. 

21.  A  merchant  has  20  hogsheads  of  tobacco,  each  weighing  9cwt. 
Iqr.  141b.,  which  he  wishes  to  put  into  an  equal  number  of  small  and 
large  boxes,  the  former  to  hold  2|lb.  and  the  latter  3  times  as  much  ; 
what  number  of  each  must  we  have ^  A.   1,878  boxes. 

22.  How  many  sheets  of  paper  will  it  take  to  make  an  18mo.  book 
(VII.  80.)  which  shall  contain  288  pages  ( =  144  leaves  1)  A.  8  sheets. 
How  many  quires  to  print  an  edition  of  only  96  copies  ]  A.  32  quires. 
How  many  reams  to  print  an  edition  of  2,400  copies  ?   A.  40  reams. 

23.  At  83.50  per  ream,  what  will  be  the  expense  of  paper  for 
l)rinting  an  edition  of  43,200  copies  of  a  12mo.  work,  to  consist  of 
192  pages,  making  the  usual  allowance  of  2  quires  of  w^aste  paper  in 
each  ream  1  A.  $2,800. 

24.  How  many  years  of  365 1  days  in  49,000  hours'? 

Note.— In  214f  days,  the  §  of  a  day  is  of  course  f  of  24  hours  = 
18  hours,  which  added  to  16  hours,  the  first  remainder  =^34h.  =  Id. 
lOh.    Add  the  1  day  to  the  214  days.  A.  5Y.  215d.  lOh. 

25.  "A  gentleman  in  Buffalo  has  just  (Feb.  1838)  sold  all  his  real 
estate  for  $130,000,  payable  in  instalments  at  the  rate  of  1  dollar  an 
hour."  What  period  of  time  has  the  purchaser  allowed  him  for  the 
payment  of  the  debt,  reckoning  365^  days  to  the  year  1 

^  A.  14Y.  303d.  4h. 


FRACTIONS. 

GENERAL  PRINCIPLES. 

LI.  1.  When  two  numbers  are  written,  one  above  the  other  wdth 
a  line  betw^een  them,  they  mean  as  follows  :— 

i  (1-half)        means  1  of  the  2  equal  parts  of  a  unit  or  any  thing- 

i  (1-third)      means  1  of  the  3  equal  parts  of  a  unit  or  any  thing. 

f  (2-thirds)    means  2  of  the  3  equal  parts  of  a  unit  or  any  thing. 

I  (1-fourth)   means  1  of  the  4  equal  parts  of  a  unit  or  any  thing. 

f  (3-fourths)  means  3  of  the  4  equal  parts  of  a  unit  or  any  thing. 

\  (1-fifth)       means  1  of  the  5  equal  parts  of  a  unit  or  any  thing. 

f  (2-fifths)     means  2  of  the  5  equal  parts  of  a  unit  or  any  thing. 

I  (5-fifths)  means  5  of  the  5  equal  parts,  that  is,  the  whole  ot  any 
thing,  and  so  on  in  respect  to  any  numbers  whatever. 

2.  Then  f ,  or  f ,  or  j,  or  |,  or  I  &c.  are  each  equal  to  1  unit. 

LI.  Q.  What  is  meant  by  1  f,  |,  &c.?  1.  What  by  f ,  or  f ,  |,  &c.?  2. 
10 


110  ARITHMETIC. 

3.  These  expressions  are  called  Fractions  (from  the  hsLtin  f radio 
signifying  broken,)  because  they  stand  for  numbers  broken  or  divided 
into  parts. 

4.  The  whole  unit  or  thing,  of  which  fractions  are  broken  parts,  is 
called  an  Integer  (a  Latin  word  signifying  whole,)  in  order  to  dis- 
tinguish it  from  fractions. 

5.  Fractions  then  are  the  expressions  for  one  or  more  equal  parts  of 
a  unit  or  whole  number,  called  an  integer. 

6.  The  number  below  the  line,  which  shows  into  how  many  equal 
parts  the  unit  or  integer  is  divided,  is  called  a  denominator'  ;  because 
it  gives  the  name  or  denomination  to  the  fraction  ;  as,  halves,  thirds^ 
&c. 

7.  The  number  above  the  line,  which  shows  the  number  of  parts 
meant,  is,  for  that  reason,  called  the  numerator.'  The  Numerator 
and  Denominator  are  called  the  Terms  of  the  Fraction. 

8.  Thus,  in  -J,  f,  |,  |,  the  upper  terms,  1,  3,  5  and  7  are  the 
numerators,  and  the  lower  terms,  2,  4,  6  and  8,  are  the  denominators. 

LII.  1.  Since  the  denominator  represents  all  the  parts  of  the 
integer,  therefore, — 

2.  Ifive  multiply  the  value  of  a  single  part  by  the  denominator,  the 
product  will  he  the  entire  value  of  the  integer. 

3.  When  j^  of  a  bushel  of  rye  costs  12  cents,  what  will  y^  or  1 
bushel  cost?  A.  SI. 20. 

4.  If  ^  of  a  vessel  be  valued  at  85,000,  what  is  the  value  of  the 
whole  vessel?  A.  §25,000. 

5.  What  is  that  number  of  which  36  is  ^V  A..  1,080. 

6.  29  is  is  of  what  number  1  A.  1,450. 

7.  75  is  iV  of  what  number  ]  A.  3,000. 

8.  When  f  of  a  cask  of  wine  sells  for  $45,  what  is  the  whole  cask 
worth  at  that  rate  1  Find  the  value  of  \  first,  by  dividing  45  by  3, 
then  multiply  the  result  by  4 1  A.  $60- 

9.  21  is  3%  of  what  number  ]  The  result  will  be  the  same,  if  we 
multiply  by  12  first  and  divide  by  3  afterwards ;  thus,  24  X  12-^3 =96. 

A.  96. 

10.  Hence  if  we  multiply  the  value  of  any  fraction  by  its  denom- 

Q.  What  are  such  expressions  called  and  why  ?  3.  What  then  are  Fractions  ? 
5.  What  is  an  Integer  and  whence  its  name  ?  4.  What  is  the  figure  below  the 
line  called,  and  why  ?  6.  What,  the  figure  above  the  line,  and  why  ?  7.  What  do 
both  the  numerator  and  denominator  form?  7.  Which  are  the  numerators  and 
denominators  in  ^  and  f  ?  8. 

LII.  Q.  How  may  the  value  of  any  integer  be  ascertained  from  having  its 
fractional  part  given?  2.  Why  so  ?  1.  When  you  pay  3  dollars  for  j  of  a  ton  of 
hay,  what  would  be  the  price  of  a  whole  ton  ?  When  ^  of  a  hogshead  of  molasses 
costs  12  dollars,  what  is  the  price  of  a  whole  hogshead  ?  What  is  the  rule  for 
it?  10. 

1  Demominator,  [L.  denomino.']  He  that  names. 

2  NUMEBATOR,  [L.  numero.}  One  that  numbers. 


FRACTIONS.  Ill 

inator,  and  divide  the  result  by  its  numerator ,  the  product  will  be  the 
entire  value  of  the  integer. 

11.  If  ^  of  a  ship's  cargo  be  valued  at  810,000,  what  is  the  value 
of  the  entire  cargo  ?  A.  $15,714f . 

12.  509  is  Y%  of  what  number  ?  A.  l,323f . 

13.  The  fractional  remainders,  f  and  f  above,  are,  properly  speak- 
ing, unexecuted  divisions ;  hence  fractions  are  said  to  have  originated 
in  this  manner  from  Division. 

14.  815  is  fl  of  what  number?  A.  2,051^- 

15.  940  is  f ^  of  what  number  ?  A.  l,019f|. 

LIII.  1.  How  many  halves  are  there  in  17  dollars?  Since  2- 
halves  are  equal  to  1  dollar,  there  are  2  times  as  many  halves  as  there 
are  dollars.  A.  34  halves =^*. 

2.  Hence  multiplying  any  whole  number  by  a  given  denominator, 
shoivs  hoio  many  parts  are  to  be  taken  for  the  numerator. 

3.  How  many  dollars  are  \^  of  a  dollar  ?  Evidently  as  many  dol- 
lars as  there  are  times  2  in  34,  for  2-halves  make  1  dollar. 

A.  17  dollars. 

4.  Hence  dividing  the  numerator  by  the  denominator,  shows  what 
whole  number  is  contained  in  the  fraction. 

5.  How  many  fourths  or  quarters  in  $5  ?  Sixths  in  118  bushels  % 
Sevenths  in  395  barrels  ? 

6.  How  many  dollars  in  ^^  of  a  dollar  ?  Bushels  in  -^^  of  a  bushel  ? 
Barrels  in  ^-V-  of  a  barrel  ? 

7.  Change  10  to  a  fraction  whose  denominator  shall  be  8.  A:  ^^. 
How  many  units  in  ^-^1  A.  10. 

8.  Change  625  to  a  fraction  whose  denominator  shall  be  1.  How 
many  units  are  there  in  ^^  1 

9.  Since  no  number  is  affected  by  multiplying  or  dividing  it  by  1, 
therefore, — 

10.  Any  whole  number  becomes  a  fraction  by  simply  writing  I,  for 
its  denominator. 

11.  What  fraction,  that  has  17  for  a  denominator,  is  equal  to  365 1 
Or  to  415?  A.  ■^^^;  ^^K 

12.  When  1  pound  of  butter  costs  jV  of  a  dollar,  how  many  pounds 
may  be  bought  for  $1?     For  $365?  A.  lOIb  ;  3,0501b. 

13.  When  1  gallon  of  molasses  costs  ^  of  a  dollar,  what  will  be 
thecostof  5gal.?     Of20gal.?     Of  1  tierce?        A.  $1 ;  S4;  $8f. 

14.  How  many  furlongs  are  equal  to  ^%^  fur.  ?  A.  61|-  furlongs. 

15.  Hence  the  value  of  any  fraction,  is  the  quotient  arising  from 
dividing  the  numerator  by  the  denominator. 

Q.  20  is  I  of  what  number?  How  did  fractions  originate?  13. 

LIII.  Q.  How  many  halves  are  there  in  17  dollars,  and  why?  1.  How  is  it 
ascertained?  2.  In  10  minutes  how  many  fourths? — fifths? — sixths?  How  many 
dollars  are  there  in  3_4  of  a  dollar,  and  why  ?  3.  What  is  the  inference  ?  4.  How 
many  furlongs  in  Q^"6f  a  furlong?— in  ^V  "f"  a  furlong?  How  does  any  whole 
number  become  a  fraction?  10.  Why  so?  9.  Give  an  example. 


112  ARITHMETIC. 

16.  Wliat  is  the  value  of  ^^  of  a  bushel  1       A.  135^  bushels. 

17.  What  is  the  value  of  V  ^     Of  ''f '  ^  ^-  4 ;  415. 

18.  Fractions  it  seems  are  proper  indications  of  Division,  in  ivhich 
the  numerator  is  the  dividend,  the  denominator  the  divisor,  and  the 
quotient  the  value  of  the  fraction,  xxviii.  17. 

19.  What  is  the  value  of  that  fraction,  which  may  be  formed  by 
the  divisor  21  and  the  dividend  6,170?  A.  293||. 

20.  What  is  the  value  of  4,500  divided  by  91  ?  A.  49f|. 

21.  When  the  denominator  is  18  and  the  value  25,  what  is  the 
numerator]  A.  450. 

22.  When  the  numerator  is  3,645  and  the  value  81,  what  is  the 
denominator]  A.  45. 

23.  When,  however,  the  dividend  is  less  than  the  divisor,  the  quo- 
tient is  the  fraction  formed  byivriting  the  divisor  under  the  dividend. 

24.  Divide  $1  equally  among  4  persons.  A.  %\  apiece. 

25.  Divide  3  by  5.     4  by  9.     723  by  901.  ^-  §  ;  I ;  wt- 

26.  If  20  bushels  of  wheat  be  divided  equally  among  23  poor  per- 
sons, what  will  be  each  one's  part  T  ^4.  f  |  of  a  bushel- 

LIV.  1 .  It  is  plain  that  every  number  is  divisible  into  as  many- 
equal  parts  as  it  contains  units, — 

2.  Thus  8=^8  units  or  8  equal  parts :  so  5=5  units  or  5  equal  parts. 

3.  Hence  if  it  be  asked,  what  part  of  8  is  5,  we  say  | ;  because 
this  means,  as  we  have  seen,  5  of  8  equal  parts. 

4.  What  part  of  7  is  3  ]  ^.  3  of  7  parts,  that  is,  f 

5.  Hence  every  number,  whicJi  is  to  become  a  part  of  another,  is 
properly  the  numerator  of  that  Fraction,  ivhose  denominator  is  that 
other  number. 

6.  What  part  of  19  is  151  A.  |f-  What  part  of  10  is  5?  A.  yV 
What  part  of  5  is  101  A.  \^=2. 

7.  When  hay  sells  for  S 10  a  load,  how  many  loads  may  be  bought 
for  $10  1     For  $71  ^.1  load ;  y^^  of  a  load. 

8.  What  part  of  120  is  401  A-  ^. 

9.  What  part  of  40  is  120  1  A.  3. 

10.  What  part  of  3  is  500 1  A.  166f .  - 

11.  What  part  of  500  is  31  A.  sh- 

12.  Suppose  you  owe  $23  and  pay  $15,  what  part  of  the  debt  do 
you  pay,  and  what  part  do  you  still  owe  1  A.  $1^;  $2^. 

LV.   1.  Since  fractions,  having  different  numerators  but  the  same 

Q.  What  appears  to  be  the  value  of  a  fraction  ?  15.  What  is  the  value  of  V  ? — 
of  '^^  ? — of  \~^  ?  Of  what  are  fractions  proper  indications  ?  18.  In  what  par- 
ticulars do  fractions  correspond  to  Division?  18. 

LIV.  Q.  Why  is  8,  for  instance,  said  to  have  that  number  of  equal  parts  ?  1. 
What  part  of  8  is  5,  and  why?  3.  How  do  you  find  what  part  one  number  is  of 
another?  5.  What  part  of  20  is  3  ?— is  8  ?— is  40  ?  Suppose  that  you  owe  60  dol- 
lars, and  pay  20  dollars ;  what  partof  the  debt  do  you  pay,  and  what  part  do  you 
still  owe  ? 

LV.  Q.  How  may  fractions  be  added  and  subtracted  ?  2.  What  is  the  reason 
for  the  process?  1. 


FRACTIONS.  113 

denominator,  express  parts,  each  of  equal  magnitude  or  value,  it 
follows, — 

2.  That  the  operations  of  addition  and  subtraction  of  fractions 
having  the  same  denominator^  may  be  performed  by  means  of  the 
numerators  alone,  in  the  same  manner  as  ivhole  numbers. 

3.  John  has  ^\  of  a  dollar,  Rufus  y^j,  and  Thomas  y\ ;  how  many 
twelfths  have  they  alH  A.  ^. 

4.  Add  together  0%,  /j*  /jj  and  /g.  A.  ft=l. 

5.  Suppose  a  man  owns  \^  of  a  sloop  and  sells  5I  of  it ;  what  part 
does  he  still  own  T  A.  3^. 

6.  How  much  does  \ll  from  ||§  leave  1  A.  ,Vf . 

7.  A  boy  having  Si,  paid  away  -^^  of  it ;  how  many  sixteenths  had 
he  left  I    ($l=|f.)  A.  -H. 

8.  Subtract  ^f  from  1  unit  (=§|.)      ^  A.  ||. 

9.  What  is  the  sum  of  g^f  and  f|'?         •  A.  \. 

10.  Add  together  Sf,  $i,  Si  and  Si  A.  SV=2. 

11.  Add  together  |^,  |i  ||,  /«,  and  |f.  JL.  3. 

12.  What  is  the  sum  of  f ,  VS  f ,  f  1  f  and  ^  ?  A.  5|. 

13.  How  much  less  than  1  is  ii^%%%  ?  A.  ^lUm- 

LVI.  1.  Since  the  greater  the  number  of  parts  used,  the  greater 
must  be  the  value  of  the  fraction,  and  the  reverse,  therefore, — 

2.  A  fraction  is  as  many  times  greater,  as  its  numerator  is  made 
greater;  and  as  many  times  smaller,  as  its  numerator  is  made 
smaller. 

3.  Hence  multiplying  the  numerator  multiplies  a  fraction,  and 
dividing  the  numerator  divides  a  fraction. 

4.  If  1  yard  of  ribbon  costs  y\  of  a  dollar,  what  will  5  yards  cost  ? 

3  x5=-^.    A.  ^. 

5.  Multiply  2^3  by  2  ;  by  3  ;  by  4.  '     A.  l| ;  if ;  f  f . 

6.  If  5  yard^  of  ribbon  cost  yl  of  a  dollar,  what  will  1  yard  cost? 

A.  yi-^5=^  of  a  dollar. 

7.  Divide  1|^  by  50  ;  |^|  by  270 ;  A.^,;  y^. 

8.  Multiply  3^7  by  45;  by  90;  by  110.  A   1 ;  2  ;  2\\^. 

9.  If  a  horse  consume  in  1  day  o^f^  of  a  ton  of  hay,  how  much 
would  he  consume  in  1  week  1     In  1  month  1     In  1  year  ? 

A         1  3  3  np  .      570  rp  .     q  9  3  5  np 

-^-  ^offiT^'  2oTnr^-'  •'2offo^- 

10.  How  many  times  greater  is  -^-^  than  y^y  ^  90-^15=6  times, 
the  answer ;  for  from  No.  1  and  2  above  it  follows,- — 

Q.  What  is  the  sum  of  ^,  -3^^  and  -^^  l  What  is  the  diflference  between 
j?y  and  the  sum  of  ^-^  and*  -^^  f 

LVI.  Q.  On  what  does  the  value  of  a  fraction  depend?  1.  How  then  may 
a  fraction  be  made  greater  or  smaller?  2.  What  is  the  inference  in  respect  to 
multiplying  or  dividing  a  fraction  ?  3.  Divide  ||  by  8 ;  by  24.  Multiply  ^ 
by  5 ;  by  6;  by  15.  If  ||  of  a  dollar  will  buy  5  dozen  of  eggs,  what  is  a  single 
dozen  worth?  How  many  times  is  X  contained  in  |4^1  Why  divide  the  60 
by  5?  3. 

10* 


114  ARITHMETIC, 

11.  That  when  two  fractions  have  the  same  denominators^  one  is 
as  many  times  greater  than  the  other,  as  the  numerator  of  the  one  is 
contained  times  in  the  numerator  of  the  other. 

12.  Divide  W  by  -i^ ;  mi  by  AW •  ^-  50 ;  5. 

13.  Suppose  you  plant  ^  of  a  bushel  of  corn  on  an  acre,  and  that 
it  yield  400  times  that  quantity ;  how  many  bushels  will  you  gather  ? 

A.  50  bushels. 

LVII.  1.  Since  the  greater  the  number  of  parts  into  which  any 
thing  is  divided,  the  smaller  each  part  must  be,  and  the  reverse, 
therefore, — 

2.  A  fraction  is  as  many  times  greater  as  its  denominator  is  made 
smaller,  and  as  many  times  smaller  as  its  denominator  is  made  greater. 

3.  Hence  dividing  the  denominator  multiplies  the  fraction,  and 
multiplying  the  denominator  divides  the  fraction. 

4.  If  a  father  divides  |^  of  a  dollar  equally  between  his  2  sons,  what 
part  of  a  dollar  will  each  have  1^x2=^.  A.  $|. 

5.  Divide  |  by  2 ;  by  4  ;  by  6  ;  "by  8 ;  by  11. 

A     1  .    1  .   JL .     1    .     1 

-^*   T  >    7  »    1  2  '    TS"'    22* 

6.  When  the  price  of  cotton  cloth  is  ^^  of  a  dollar  a  yard,  what 
will  be  the  cost  of  4  yards  ]     Of  8  yards?  Of  16  yards'?     (-rff-^4=f> 

A.  $f  ;  Sg'  >  $!• 

7.  Multiply  ^g-  by  8;  by  320  ;  by  960.  ^.  _|^ ;  1  .  3. 

8.  Hence  suppressing  the  denominator,  multiplies  the  fraction  by 
that  number. 

9.  Multiply  ^1^  by  780  ;  |-f  by  96.  A.  315  ;  42. 

10.  Divide  H  by  14  ;  by  21 ;  by  45.  A.  ^^^  ;  Us  ;  rrh- 

1 1 .  When  your  board  for  \  of  a  month  costs  V  of  a  dollar,  what 
would  the  board  for  1  month  cost?  A.  klb. 

12.  We  see  from  the  above,  that  when  several  numerators  are 
alike,  the  greatest  fraction  has  the  smallest  denominator,  and  the 
reverse, — 

13.  Thus  y2^  is  less  than  y%,  or  1^,  &c.;  so  ^^-^  is  many  times 
smaller  than  f . 

14.  Again,  when  the  denominators  are  alike,  the  greatest  fraction 
has  the  greatest  numerator, — 

15.  Thus  I  is  greater  than  ^  or  I,  &c.;  so  y»^  is  99  times  greater 
than  Y^. 

16.  We  have  now  two  ways  for  multiplying  a  fraction,  and  two 
ways  for  dividing  it,  viz  : — 

17.  A  fraction  is  multiplied  by  multiplying  its  numerator,  or  by 
dividing  its  denominator. 


LVII 


Its 


-iVII,     Q.  What  effect  is  produced  on  a  fraction  by  increasing  or  decreasing 
denominator?  2.     Why  has  it  this  effect?  ].     How  then  may  a  fraction  be 


FRACTIONS.  115 

19.  A  fraction  is  divided  by  dividing  its  numerator,  or  by  multiply- 
ing- its  denominator. 

19.  Multiply  4^0  by  5  both  ways.  A.  Ui  =  l ;  U=l. 

20.  Divide  VV'  by  10  both  ways.  A.  \^=l ;  |tJ=l. 

LVIII.  1.  Since  multiplying  the  denominator  has  an  opposite 
effect  from  multiplying  the  numerator,  and  dividing  the  denominator 
an  opposite  effect  from  dividing  the  numerator,  therefore, — 

2.  When  both  the  numerator  and  denominator  are  either  multiplied 
or  divided  by  the  same  numhcr,  these  operations  must  compensate  or 
balance  each  other;  that  is,  have  no  effect  on  the  value  of  the  fraction. 

3.  Find  how  many  half  dollars  are  equal  to  |  of  a  dollar,  by  dividing 
each  term  by  4.  A.  ^\. 

4.  Find  how  many  eighths  of  a  dollar  are  equal  to  |,  by  multiplying 
each  term  by  4.  A.  $|. 

5.  That  j  is  equal  to  |  is  obvious  from  its  meaning  ,4  of  8  parts, 
which  are  of  course,  \  of  the  whole. 

6.  Find  what  other  fractions  are  equal  to  /o,  by  multiplying  each 
term  by  3 ;— by  5  ;— by  8.  "     A.  fl ;  |^  ;  U- 

7.  Find  what  other  fractions  are  equal  to  f^-j^,  by  dividing  each  term 
by  5  ;— by  8  ;— by  120.  -A-  41 ;  ^r,  f  • 

8.  Change  f  n^  to  eighths  by  dividing  each  term  by  any  number  that 
will  make  the  denominator  8.  A.  |. 

9.  Change  |  to  fortieths  by  multiplying  each  term  by  any  number 
that  will  do  it. 

10.  Change  4  to. fiftieths,  and  |^  to  fifths. 

11.  Change  f  to  ninetieths,  and  f  g-  to  thirds. 

12.  John  has  83  j|,  Ilufus  Sjf  y,  and  Harry  $-j%  ;  how  many  thirds 
of  a  dollar  has  each  1  A%\. 

13.  Reduce  ||  to  tenths,  and  -^^  to  |.  Because  the  terms  in  \ 
cannot  be  divided  again  by  any  number  greater  than  1,  without  a  re- 
mainder, the  fraction  is  said  to  be  in  its  loivest  or  most  simple  terms » 
and  the  terms  themselves  to  be  prime  to  each  other. 

14.  Reduce  -j^,  h^,  and  |f^  to  their  lowest  terms.     J..  | ;  ^;  |^. 

15.  Reduce  ||^|  to  200ths ;  \ll  to  20ths  ;  ^l  to  fourths.    A.  |. 

16.  Reduce  £j^^^q  to  its  lowest  terms,  by  dividing  by  any  number 
that  will  divide  both  terms  without  a  remainder,  and  these  quotients 
again  as  before,  and  so  on  till  the  terms  become  prime  to  each  other. 

A.  £1 

17.  Reverse  the  last  process  and  change  £j  to  1200ths. 

A.  £^^^ 


Q.  Which  is  the  greater  fraction,  -?j.  or  -^ ;  ^  or  ^  1  What  are  the  two 
ways  for  multiplying  or  dividing  a  fraction?  17,  18. 

LVIII.  Q.  What  operations  on  fractions  will  produce  opposite  effects  ?  1 . 
How  may  these  effects  be  counteracted?  2.  What  is  the  proof  that  |  is  equal 
to  It  5.  When  are  fractions  reduced  to  their  lowest  terms?  13.  How  are 
they  reduced  to  such  terms  ?  16.     Reduce  ^  and  ^°^  to  their  lowest  terms. 


116  ARITHMETIC, 

18.  Reduce  as  above,  ?f  ^yd.  to  its  lowest  terms.  A.  ^yd. 

19.  Suppose  one  man  buys  yVo  of  a  barrel  oF  flour,  another  |^f  of  a 
barrel,  a  third  y^^Vbl.,  a  fourth  4|^bl.,  a  fifth  jVAW.,  and  the  sixth 
T3wW.;  what  part  of  a  barrel  has  each  man"?  A.  5-. 

20.  What  is  the  greatest  number  that  will  divide  without  a  re- 
mainder both  terms  in  |f  ^,  and  what  are  the  most  simple  terms  of 
this  fraction  ?  A.  60;  f. 

21.  The  60  in  the  last  example  is  called  the  greatest  common 
divisor  of  the  terms  of  the  fraction,  and  by  means  of  it  the  fraction 
is  reduced  at  once  to  its  most  simple  terms. 

22.  Hence  the  importance  of  a  rule,  by  which  the  greatest  com- 
mon divisor  may  in  all  cases  be  easily  ascertained. 

LIX.  To  find  the  greatest  common  divisor,  or  as  it  is  sometimes 
called,  the  greatest  common  measure,  of  two  or  more  numbers. 

1 .  When  a  number  greater  than  1  will  divide  another  without  a 
remainder,  it  is  called  a  measicre  or  even  divisor  of  that  number. 

2.  Find  by  trial,  all  the  even  divisors  or  measures  of  12  and  16. 

A.  Of  12:  2,  3,  4,  12.     Of  16:  2,  4,  8,  16. 

3.  When  a  number  greater  than  1  will  divide  two  or  more  numbers 
without  a  remainder,  it  is  called  their  common  measure  or  common 
divisor. 

4.  Find  by  trial,  all  the  common  divisors  of  12  and  16.   A.  2:  4. 

5.  The  greatest  number  that  will  divide  in  this  manner  two  or 
more  numbers,  is  called  their  greatest  common  divisor. 

6.  Find  by  trial,  the  greatest  common  divisor  of*  24  and  32.  Of 
175  and  252.  A.  8  ;  7. 

7.  Find  the  greatest  common  divisor  of  240  and  480.  Of  12,  36, 
and  48.  A.  240;  12. 

8.  In  these  examples  the  smallest  number  is  the  divisor  sought ; 
whether  this  be  the  case  with  any  two  numbers  is  easily  ascertained 
by  dividing  the  greater  by  the  less  ;  thus,  taking  125  and  625 ; — 

125)625(5  ^-  Here  125  is  an  even  divisor 

6  2  5  of  625,  and  because  125  can  have 

p      r    1   Q  K  N  1   Q  K— 1  ^®  greater  even  divisor  than  itself; 

12  5)62  5=5  therefore  125  is  the  greatest  com. 

—  divisorof  125and625.  ^.   125. 

10.  Find  the  greatest  common  divisor  of  375  and  2,250.     Of  1,817 

and  21,804.  A.  375;  1,817. 

LIX.  Q.  What  is  a  measure  of  a  number?  1.  What  is  meant  by  a  common 
measure  or  a  common  divisor?  3.  What  are  the  common  divisore  of  12  and 
16?  4.  What  is  meant  by  the  greatest  common  divisor?  5.  What  is  the  great- 
est common  divisor  of  125  and  625?  How  is  it  ascertained?  8.  Why  is  12  a 
common  divisor  of  72  and  84,  rather  than  the  smaller  of  the  given  numbers  ?  12. 
But  why  is  12  the  greatest  common  divisor  of  72  and  84?  13,  What  inference 
is  drawn  in  respect  to  the  common  divisor  of  two  numbers,  and  the  difference 
between  these  numbers  ?  14.  What  is  the  general  rule?  16.  When  have  num 
bers  no  common  divisor  ?  17. 


FRACTIONS.  117 

1 1 .  Wliat  is  the  greatest  common  divisor  of  72  and  84 1 

7  2)84(1  12.  Here  the  smaller  number  is  not  the 

7  2  divisor  sought,  for  12  remains  in  dividing ; 

12)72(6  but  since  72  is  exactly  divisible  by  12  ;  84 

7  2  must  be  so  also,  for  84  being  12  more  than 

Proof    12)72  ==6  ''^'  "^^^^  contain  12  exactly  once  more 

'  12)84^7  t'lan  72. 

13.  The  number  12  then  is  a  common  divisor  of  72  and  84,  and 
since  84  is  only  12  m.ore  than  72,  it  is  plain  that  no  number  greater 
than  12,  that  will  divide  72  even,  can  divide  84  even  also,  therefore 
12  is  the  greatest  common  divisor  of  72  and  84. 

14.  We  learn  from  this  illustration,  that  the  greatest  common 
divisor  of  two  numbers,  never  exceeds  their  difference. 

15.  Since  the  same  reasoning  which  is  employed  in  example  11 
and  12  would  apply  to  any  number  of  successive  divisions  ;  therefore, 
we  have  the  following, — 

GENERAL  RULE. 

16.  Divide  the  greater  number  by  the  less,  and  that  divisor  hy  the 
remainder,  and  so  on ;  always  dividing  the  last  divisor  by  the  last 
remainder,  till  nothing- remains  ;  the  last  divisor  is  the  greatest  com- 
mon divisor  required. 

17.  When  the  last  divisor  is  1,  the  given  numbers  are  prime  to 
each  other,  and  therefore  have  no  common  divisor. 

18.  Find  the  greatest  common  divisor  of  495  and  585. 

4  9  5)585(1  Proof 

4  9  5  45)495  =  11 

90)495(5  45)58  5  =  13 

4  5^^ 10.  The  last  divisor  is  45,  and 

Gr.  com.  divi.         1[  5  )  9  0  (  2  it  leaves  no  remainder ;  therefore 

9  0  45  is  the  greatest  com.  divisor. 
'  Answer,  45. 

20.  What  is  the  greatest  com.  divisor  of  356  and  788?       A.  4. 

21.  Whatisthegreatestcom.  divisor  of  1,190  and  2,225?    A.  5. 

22.  What  is  the  greatest  common  divisor  of  3,760  and  9,024  \ 

'A.  752. 

23.  When  there  are  more  than  two  numbers — First  find  the  great- 
est common  divisor  of  any  two  of  them,  then  of  that  common  divisor 
and  a  third,  and  so  on ;  the  last  common  divisor  will  be  the  greatest 
common  divisor  of  all  the  numhers. 

24.  What  is  the  greatest  common  divisor  of  54,  126  and  1801 
The  common  divisor  of  54  and  126  is  18,  and  of  18  and  186  is  6. 

A.  6. 

25.  What  is  the  greatest  common  divisor  of  3,672,  5,832  and  1,0441 

A    36. 

26.  Have  183  and  719  a  common  divisor  ]  Sec  No.  17. 


118  ARITHMETIC. 

27.  What  are  the  lowest  terms  of  |^ff,  and  what  common  divisor 
will  reduce  it  to  those  terms  by  a  single  operation  1 

A.  If  °y| ;  4  divisor. 

28.  Suppose  a  piece  of  land  lies  in  the  form  of  a  triangle,  and  that 
one  side  is  85  rods  in  length,  another  75  rods,  and  the  other  20  rods; 
what  is  the  length  of  the  longest  chain  that  will  exactly  measure  each 
side  1  A.  5  rods  long. 

29.  Suppose  a  bookseller  has  an  order  from  A,  for  375  Arithmetics ; 
one  from  B,  for  450  Arithmetics,  and  another  from  C  for  525  Arith- 
metics, which  he  would  pack  in  equal  boxes,  a  certain  number  of 
which  should  just  hold  all  the  books  each  man  ordered.  What  is  the 
greatest  number  of  books  that  he  can  put  into  each  box  1 

A.  75  books. 

LX.  To  find  the  least  common  multiple  of  two  or  more  numbers ; 
a  process  used  in  reducing  fractions  to  their  least  common  denom- 
ination. 

1.  A  COMMON  MULTIPLE  of  two  or  morc  numbers  is  that  number 
which  can  be  divided  by  each  without  a  remainder. 

2.  Thus  12  is  a  common  multiple  of  3  and  4,  for  it  is  divisible  by 
each  without  a  remainder. 

3.  The  least  common  multiple  of  two  or  more  numbers,  is  the 
least  number  that  can  be  divided  by  each  without  a  remainder. 

4.  Find  by  trial,  the  least  common  multiple  of  3  and  2.      A.  6. 

5.  Find  by  trial,  the  least  common  multiple  of  4  and  6.  Of  6  and 
8.     Of  9  and  6.  A.  12;  24;  18. 

G.  When  two  or  more  numbers  are  multiplied  together,  they  are 
called /ac^or*,  and  their  product  a  compo5t7e  wwmZ»er.  xvii.   1. 

7.  Hence  every  composite  number  is  a  common  multiple  of  its  faC" 
tors,  for  it  is  of  course  divisible  by  each  factor. 

8.  Suppose  12  to  be  a  common  multiple,  and  one  of  its  factors  to 
be  3  ;  what  is  the  other  factor  ]  A.  4. 

9.  If  143  be  a  common  multiple,  and  one  of  its  factors  is  11,  what 
is  the  other  factor  ?  A.   13. 

10.  What  common  multiple  may  be  formed  by  the  factors  30  and 
71?  A.  2,130. 

1 1 .  A  Prime  Number  is  one  that  is  divisible  only  by  itself  or  unity, 
as2,  3,  5,  7,  11,  13,  17,  &c. 

12.  The  product  of  any  two  or  more  prime  numbers  or  factors^ 
is  their  least  common  multiple. 

13.  What  is  the  least  common  multiple  of  the  prime  numbers  17 
and  23? A.  391. 

LX.  Q.  What  is  meant  by  a  common  multiple?  1.  What  is  the  common 
multiple  of  3  and  4  ?  2.  What  is  meant  by  the  least  common  multiple  ?  3.  What 
is  the  least  common  multiple  of  4  and  6  ?  Of  6  and  3  ?  What  are  factors  ?  6. 
Why  is  every  composite  numl)pr  a  common  multiple  ?  7.  If  5  be  one  factor  of  a 
common  multiple  75,  what  is  the  other  factor?  What  is  a  prime  number  ?  11. 
Give  several  examples.  What  is  the  least  common  multiple  of  7  and  11  ? 


FRACTIONS.  119 

14.  Numbers  are  prime  to  each  other  when  they  have  no  common 
divisor,  lviii.   13. 

15.  The  product  of  any  two  ormoi'e  numbers  prime  to  each  other 
is  their  least  common  multiple. 

16.  What  is  the  least  common  multiple  of  2,  9  and  13  ? 

.4.  234. 

17.  What  is  the  least  common  multiple  of  the  prime  factors  2,  3, 
5,  7  and  111  A.  2,310. 

18.  What  is  the  least  common  multiple  of  the  factors  3,  4,  5  and 
7,  they  being  prime  to  each  other  ?  A.  420. 

3)120  19.  What  are  the  prime  factors  of  120  t 

2)40  20.  Here  the  divisors  and  the  last  quotient  are  all 

o  \  2  0  prime  numbers,  and  if  multiplied  together  must  make, 

^-r-Q  as  they  do  (3x2x2x5x2=)  120;  therefore  they 

comprise  all  the  prime  factors  of  120. 

^  A,  3,  2,  2,  5,  2. 


A.   : 

2,3, 

5,7,11. 

A. 

5, 

7,  11,  13. 

3,5, 

,7  and  in  Of 

A. 

2310;  5005. 

A. 

2  and  3. 

A. 

2  and  5. 

21.  Hence,  to  find  the  prune  factors  of  any  number — Divide  it 
successively  by  any  prime  number  that  will  divide  it  without  a  remain- 
der^ till  the  quotient  becomes  a  prime  number,  then  the  several  divisors 
together  with  the  last  quotient  will  become  the  prime  factors  required. 

22.  What  are  the  prime  factors  of  2310  \ 

23.  What  are  the  prime  factors  of  5005  ? 

24.  What  is  the  least  common  multiple  of  2, 
5,7,  11  and  13] 

25.  What  are  the  prime  factors  of  6 1 

26.  What  are  the  prime  factors  of  10  1 

27.  One  multiple  of  6  and  10  is  their  product =60  ;  and  GO  is  also 
a  multiple  of  all  the  prime  factors  in  both  6  and  10,  for  2  x  3  x  2  x  5 
=60. 

28.  But  60  is  not  the  least  multiple  of  6  and  10,  because  it  has  the 
factor  2  repeated,  as  2x2x3x5= 60;  therefore  we  may  drop  one  2, 
leaving  2x3  x  5  =  30,  which,  because  it  is  the  product  of  all  the  prime 
factors  that  are  necessai-y  to  produce  6  and  10,  is  the  least  common 
multiple  of  6  and  10. 

29.  When  two  numbers  have  one  superfluous  factor,  it  may  be  ex- 
cluded by  dividing  by  any  prime  number  that  will  divide  both  of  them 
without  a  remainder ;  thus,  taking  6  and  10  again, — 

2)6.10  30.  The  prime  factors  5x3x2  =  30,  the  least 

3  .       5        common  multiple  as  before. 
31.  Recollect  to  divide  by  a  prime  number,  and  to  multi-ply  both  the 
divisor  and  quotients  together  for  the  required  multiple. 

Q.  Why  ?  12.  When  are  numbers  prime  to  each  other  ?  14.  What  is  the  least 
common  multiple  of  such  numbers?  15.  What  is  the  direction  for  finding  the 
prime  factors  of  any  number?  21.  Why  is  not  60  the  least  common  multiple  of 
6  and  10  ?  28.  What  then  is  their  least  common  multiple,  and  why?  28.  How 
can  the  superfluous  factor  be  excluded  ?  29.  What  is  the  direction  for  the  process  ? 
31. 


6 

12  0. 

2  1  0 

3 

2  4    . 

4  2 

2 

8   . 

1  4 

4   . 

7 

120  ARITHMETIC. 

32.  Find  the  least  common  multiple  of  6  and  20.  A.  60. 

33.  Find  the  least  common  multiple  of  9  and  21.  A.  63. 

34.  When  tlie  numbers  contain  more  than  one  common  factor,  it 
is  plain  that  both  quotients  must  be  divided  successively,  as  long  as 
they  are  divisible  in  this  ?nanner ;  thus,  to  find  the  hast  common 
multiple  of  120  and  210. 

35.  Here  4,  one  of  the  last  quotients,  though 
not  a  prime  factor,  is  nevertheless  equal,  as  a 
multiplier,  to  its  prime  factors  2  and  2 ;  there- 
fore the  product  of  the  divisors  and  quotients 
being  in  effect  the  same  as  the  product  of  all 
the  prime  factors  necessary  to  produce  120 
and  210,  is  their  least  common  multiple. 
Then  7x4x2x3 x 5= 840  ^1. 

36.  Find  the  least  common  multiple  of  96  and  108.        A.  864 

37.  Find  the  least  common  multiple  of  48  and  216.        A.  432. 

38.  When  there  are  several  numbers  and  only  two  are  divisible  as 
above,  it  is  evident  that  the  divisor  is  not  a  factor  of  the  rest ;  these 
must  therefore  be  written  underneath  for  the  next  division,  thus, — 

39.  Observe  that  every  number,  which  is 
not  divisible  by  the  divisor,  is  written  under- 
neath with  the  quotients.  Then  2x7x3 
X  2  X  3  X  2  X  5=2,520,  which,  because  it 
contains  all  the  prime  factors  of  5,  9,  7,  40 
and  60,  is  the  least  common  multiple  of 
these  numbers. 
Then  2x7x3x2x3x2x5=2520  A. 

GENERAL    RULE. 

40.  Divide  by  any  prime  number  that  will  divide  two  or  more  of  the 
given  numbers  ivithout  a  remainder,  and  set  the  quotients,  together 
with  the  undivided  numbers  in  a  line  beneath. 

41.  Divide  the  second  line  as  before,  and  so  on  till  there  is  no  num- 
ber greater  than  1,  that  will  divide  tivo  numbers  without  a  remainder  ; 
then  the  divisors  and  numbers  in  the  last  line  being  multiplied  together^ 
will  give  the  least  common  multiple  required. 

42.  Find  the  least  common  multiple  of  5, 18,  9,  4  and  2. 

A.  180. 

43.  Find  the  least  common  multiple  of  10,  7,  11,  5  and  8. 

A.  3080. 

44.  Find  the  least  common  multiple  of  2,  5,  25, 15  and  12. 

A.  300. 

45.  Find  the  least  common  multiple  of  2, 3,  4,  5,  6, 12,  24,  30  and 
120.      

Q.  When  the  numbers  contain  more  than  one  common  factor,  how  do  you 
proceed  ?  34.  What  is  the  general  nile  ?  40,  41 .  Why  are  the  undivided  num- 
bers arranged  with  the  quotients  in  a  line  beneath?   38. 


5 

5 

.  9 

7 

40. 

60 

2 

1 

9 

7. 

8 

12 

3 

1 

9 

7. 

4 

6 

2 

1 

3 

7 

4 

2 

1 

3 

7. 

2. 

1 

CLASSIFICATION    OF    VULGAR    FRACTIONS.  121 

Note. — ^As  120  is  exactly  divisible  by  all  the  other  numbers,  each 
is  of  course  a  factor  of  120,  and  may  therefore  be  cancelled,  leaving 
120  as  the  least  common  multiple  sought.  A.  120. 

46.  Hence  to  abbreviate  the  process,  cancel  every  number  that  will 
exactly  divide  any  other  of  the  given  numbers,  and  proceed  with  those 
that  remain  as  before. 

47.  What  is  the  least  common  multiple  of  6,  8,  11,  24,  35,  5,  7, 
72  and  22 1  As  24  is  divisible  by  6  and  8,  35  by  5  and  7,  22  by  11, 
and  72  by  24,  first  cancel  6,  8,  5,  7,  11  and  24.  A.  27720 

48.  Find  the  least  common  multiple  of  2,  3,  4,  5,  6,  8,  10,  12,  16 
and  20.  A.  240. 

49.  Find  the  least  common  multiple  of  30,  15,  60,  12,  5,  20,  4,  2, 
3  and  10.  A.  60. 

50.  Suppose  a  surveyor  has  one  chain  3  rods  long,  another  4  rods, 
another  5  rods,  and  another  6  rods ;  what  is  the  shortest  distance 
that  can  be  exactly  measured  by  each  chain  ?  .A.  60  rods. 

51.  There  is  a  circular  island,  around  which  A  can  travel  in  5 
hours,  B  in  8  hours,  and  C  in  10  hours.  Now  suppose  they  all  start 
together,  and  go  the  same  way  round  it ;  how  much  time  must  elapse 
before  they  will  come  together  again  ■?  A.  40  hours. 

CLASSIFICATION    OF   VULGAR    FRACTIONS. 

so   CALLED   TO   DISTINGUISH   THEM  FROM   DECIMAL   FRACTIONS. 

LXI.  1.  A  Vulgar  or  Common  Fraction  is  one,  whose  denom- 
inator and  numerator  are  both  expressed. 

2.  A  Proper  Fraction  is  one  whose  numerator  is  less  than  its 
denominator ;  consequently  its  value  is  less  than  unity ;  as,  ^,  §,  4,  &c. 

3.  An  Improper  Fraction  is  one,  whose  numerator  is  either 
equal  to,  or  greater  than  its  denominator  ;  consequently  its  value  is 
either  equal  to,  or  greater  than  unity ;  as,  |,  |,  f ,  V,  &c. 

4.  A  Compound  Fraction  is  the  fraction  of  a  fraction,  that  is,  a 
part  of  a  part ;  as,  f  of  f :  f  of  yt,  &c. 

5.  A  Single  or  Simple  Fraction  has  but  one  numerator  and  one 
denominator,  and  is  therefore  either  Proper  or  Improper ;  as,  f  and  y . 

6.  A  Mixed  Number  is  a  whole  number  with  a  fraction  annexed ; 
as,  13f,  8f^,  &c. 

'^.  A  Complex  Fraction  is  one  that  has  a  fraction  for  its  numera- 
tor, or  for  its  denominator,  or  for  both  its  terms ;  as — 

#8     f     5h     8       6i    ^ 
7    f'l'   11    3f  '   7f' 

Q.  How  may  the  process  in  many  cases  be  shortened  ?  46.  What  reason  is 
assigned  for  it?  45.  Note. 

Q.  LXI.    What  is  a  Vulgar  Fraction  ?   1.   What  is  a  Proper  Fraction  ?   2. 
Improper  ^Fraction  ?  3.    Simple  Fraction  ?  5.    Compound  Fraction  ?  4.  Mixed 
number  ?  "6.  Complex  Fraction  ?  7.  How  many,  and  what,  appear  to  be  the  dif- 
ferent kinds  of  Vulgar  Fractions?  Give  an  example  of  each  kind? 
11 


122  ARITHMETIC.  .    ^ 

REDUCTION  OF  VULGAR  FRACTIONS. 

LXII.  1.  Reduction  of  Fractions  is  the  process  of  changing 
their  forms  without  altering  their  value. 

CASE  I. 

To  reduce  fractions  to  their  lowest  terms. 

RULE. 

1.  Divide  both  the  terms  of  the  fraction  by  any  number  that  will 
divide  them  without  a  remainder,  and  the  quotients  again  as  before, 
and  so  on,  till  no  number  greater  than  1  will  divide  them.*  Lviii.  1,2. 

2.  Or  divide  both  terms  by  their  greatest  common  divisor,    lviii.  2 1 . 

3.  Reduce  ^r^^V  ^^  its  lowest  terms  by  both  methods. 

7)       5)  " 
S)h\\=m^U=l  (Or  gr.  com.  div.  280)  yVi^^f  Ans. 

4.  Reduce  yoW  ^^  a  barrel  to  its  lowest  terms.  A.  ^. 

5.  Reduce  yyrr  ^^ ^  dollar  to  its  lowest  terms.  A.  ^. 

6.  Reduce  f^lf  of  a  tun  to  its  lowest  terms.  A.  j. 

7.  Suppose  a  merchant  has  several  remnants  of  cloth,  one  contain-^ 
ing  ^  of  a  yard,  another  ^^^,  another  -^\,  another  2WF0  j  and  another 
411 ;  how  many  quarters  of  a  yard  in  each  ?  A.  \yd. 

CASE    II. 

To  reduce  a  whole  number  to  an  improper  fraction  having  a  given 
denominator. 

RULE. 

1.  Multiply  the  whole  number  by  the  denominator  for  the  numera- 
tor.    LIII.   1,  2. 

2.  Reduce  85  to  an  improper  fraction  whose  denominator  is  8. 
Thus,  85x8=680.  A.  ^K 

3.  Reduce  2,439  to  a  fraction  whose  denominator  is  7.  ^4..  ^''-^^K 

4.  How  many  ninths  are  there  in  41 1— in  208 1— in  207 1 — in  423 1 

A       3fiP  .     1372  .     1863  .     3807 
^-    ~!ff~»      ~^~    »      ~9^  >         5      • 

5.  At  I  of  a  dollar  a  yard,  how  many  yards  of  ribbon  may  be  bought 
for  $1 1— for  $27 1— for  $3,106  ?  A.  8yd.;  216yd.;  24,848yd. 

6.  Change  295  to  halves — to  thirds — to  fourths — to  fifths — to 

_"^*U„   *^^  ^ +U«       A       590.  885.  1180.  1475.  1770.  2065 

Sixths — to  sevenths.         A.  ^- ,  -g-?    -«~  >    —5-  \    -V-  ;    —^-' 

Case  I.  Q.  How  are  fractions  reduced  to  their  lowest  terms  ?  1.  How  can 
they  be  reduced  by  one  operation  in  division  ?  2.  On  what  principle  is  the  rule 
based?  lviii.  21.     Reduce  to  their  lowest  terms  -^^,  y^^,  and  2  4^ 

Case  II.  Q.  How  is  a  whole  number  reduced  to  anlmproper  fraction  with 
a  given  denominator?  1.  What  is  the  reason  for  the  rule?  liii.  1,  2.  A  man 
having  50  dollars,  spent  it  in  as  many  days  as  that  sum  contains  fifths  ;  how 
many  days  was  he  in  spending  it?  What  fraction  may  be  formed  with  20  and 
a  denominator?   9. ^ 

*  The  following  rules  are  useful  in  finding  the  common  divisors  of  both  terms. 

A  number  ending  in  an  even  number  or  0,  is  divisible  by  2. 

A  number  ending  in  5  or  0,  is  divisible  by  5. 

A  number  ending  in  0  or  00,  &c.  is  divisible  by  10  or  100,  «&c. 

A  number  is  divisible  by  3  or  9,  when  the  sum  of  its  figures  is  divisible  by  3  or  9. 

A  number  is  divisible  by  6,  when  the  right  hand  figure  is  even,  and  the  sum  of  the 
digits  is  divisible  by  6. 

A  number  is  divisible  by  12  when  it  is  divisible  by  4  and  3. 

A  number  is  divisible  by  4  when  its  two  right-hand  digits  are  divisible  by  4. 


REDUCTION    OF    VULGAR    FRACTIONS.  123 

7.  Suppose  a  man  gives  y^g  of  a  bushel  of  rye  for  1  pound  of  sugar  ; 
how  many  pounds  of  sugar  may  be  bought  for  1  bushel  of  rye  ■?— for 
53  bushels  1— for  216  bushels  1  A.  131b.;  6891b.;  2,8081b. 

CASE    III. 

To  reduce  a  mixed  number  to  an  improper  fraction. 

1.  Multiply  the  whole  number  by  the  denominator,  and  to  the  pro^ 
duct  add  the  numerator  for  anew  numerator. 

2.  For,  whatever  number  of  parts  the  whole  number  may  make,  it 
is  plain  that  the  fraction  will  make  as  many  more  such  parts  as  are 
indicated  by  its  numerator. 

2  0  8  f        3.  How  manysevenths  are  there  m  208^  weeks  T 

7  In  multiplying  208  by  7-sevenths,  add  in  the 

14  5  9         3-sevenths  thus :  7  times  8  are  56  and  3  are  59, 


268 


4.  Reduce  USy^y  to  an  improper  fraction.  A.  -fj-^ 

5.  Reduce  986^  to  an  improper  fraction.  A.    -*ifT^'. 
6    Change  210^  to  fifths ;  342^  to  sixths ;  425f  to  thirds ;  305f 

.  ^1  J1053.     2057  .     L2U  .     2139 

to  sevenths.  ^-    -tt  »    "s~  »      3 '   »      f    • 

7.  If  a  horse  eat  1  bushel  of  oats  in  ^  of  a  week,  how  many  bushels 

will  he  eat  in  1  week?— in  If  weeks  1— in  4f  weeks'!— in  219? 

weeks?  A.  7;  12;  33;  1,535. 

8    At  1  of  a  dollar  a  yard,  how  many  yards  of  cloth  may  be  bought 

for  618|  dollars?  .  .  «.  tv  ^'  ^'^^^  ^^'^^' 

C  A  S  E  I V  . 

To  reduce  an  improper  fraction  to  a  whole  or  mixed  number. 

RULE. 

1.  Divide  the  numerator  by  the  denominator,     liii.    15. 

2.  A  man  by  saving  jt  of  a  dollar  a  day,  saved  in  33  days  f| ;  how 
many  dollars  is  that  1  A.  $2^^- 

3.  Reduce  Vo^  to  a  mixed  number.  A.  44y|. 

4.  Reduce  "^V  to  a  mixed  number.  A.  4|f . 

5.  Reduce  W  to  a  whole  number.  A.  36. 

6.  If  a  man  spend  daily  |  of  a  dollar,  how  much  will  he  spend  m  8 
days?— in  365  days?  A.^  $1 ;  S45|. 

7.  In  V/  of  an  hour,  how  many  hours  ?        A.  5^-^=5|  hours. 

8.  If  a  steamboat  sail  1  mile  in  yV  of  an  hour,  how  long  will  it  be 
in  performing  a  trip  of  205  miles?  A.  13|  hours. 

CASE    V. 

To  reduce  a  compound  fraction  to  a  simple  one. 

RULE. 

1.  Multiply  the  numerators  together  for  a  new  numerator,  and  the 
denominators  together  for  a  new  denominator. 

Case  III.  Q.  How  is  a  mixed  number  reduced  to  an  improper  fraction?  1. 
Why  add  in  the  numerator?  2.  How  is  208|  reduced  to  an  improper  fraction  ?  3. 
Suppose  the  toll  at  a  certain  gate  is  )  of  a  dollar  for  a  sulkey ;  how  many  times 
can  it  pass  for  7^  dollars  ? 

Case  IV.  Q.  How  is  an  improper  fraction  reduced  to  a  whole  or  mixed 
number?  1.    Why  divide  by  the  denominator?  liii.  15. 


124 


ARITHMETIC. 


2.  A  man,  owning  |  of  a  vessel,  sold  |  of  his  share  ;  what  part  of 
the  vessel  did  he  sell  1 

3.  He  sold  ^  of  ^  of  the  whole  vessel.  To  get  ^  of  any  number 
we  divide  by  5  ;  but  to  divide  a  fraction  we  may  (by  lvii.  3.)  multiply 
the  denominator,  thus  :  ^xs^ws  5  then  ^  of  ^  would  be  4  times  as 
much,  which  (by  lvi.  3.)  is  ^%--^U-  Therefore,  f  of  f=f  x^  = 
^j,  Answer. 

4.  Reduce  -j^  of  ||f  to  a  simple  fraction.  A.  Htt- 

5.  Reduce  f  of  |f  f  to  a  simple  fraction.  A.  yWj. 

6.  A,  having  f  of  a  grist  mill,  sold  f  of  his  part  to  B,  who  sold  | 
of  his  part  to  C  ;  what  part  of  the  mill  does  C  own  1  A.  |. 

7.  How  much  is  A  of  4  of  1  of  j\  ]  A.  ^%\=A\. 

8.  Howmuchisf  off  of  5  off?  A.  ff 

9.  When  a  whole  or  mixed  number  occurs,  reduce  it  first  to  an 
improper  fraction,  then  proceed  as  before. 

10.  What  is  I  of  20?     (20=  V)  J..  i^^=V=12|. 

11.  What  is  i  of  I  of  f  of  1,000  ?  A.  208^ 

12.  Whatisf  of  401  yards?     (40|=^i^.)  A.  16}  yards. 

13.  What  is  I  of  f  of  4  of  20|  gallons  %  A.  5||  gallons. 

14.  A  having  208|  hogsheads  of  molasses,  sold  f  of  it  to  B,  who 
sold  f  of  what  he  bought  to  C,  who  sold  }  of  what  he  bought  to  D. 
How  many  gallons  did  each  purchaser  buy  ? 

A.  B  125}  gal.;  C  83^  gal;  D  20^  gal. 

15.  When  any  two  opposite  terms  have  a  common  divisor,  use  their 
quotients  in  their  stead;  and  when  they  are  alike,  cancel  both,  which 
is  called  cancelling  equal  terms. 

16.  For,  the  effect  is  that  of  dividing  both  terms  of  the  product  by 
the  same  number,  which  (by  lviii.  2.)  does  not  alter  the  value. 

17.  What  is  I  of  I?     Cancel  the  5s.  A.  |.      ' 

18.  What  is  f  of  f  of  f  f  f  of  a  hogshead  ?  A.  if  |f  hhd. 

19.  What  is  f  of  f  of  f  of  5\  pints  ?  A.  l4  pints. 

20.  What  is  \\  of  yVtt  ^  The  greatest  common  divisor  of  38  and 
67  is  19  ;  therefore  ^i  of  jVq  =  V  x  ih-  ^-  ■^^^ 

91      What  i«  3  18  nf  24  5  nf  i-Sflll       (z=ly^l^M.\  A       2_fl  4  5  6 

«Jl.     vvnailS     35    01  3T?  01   3yj-y  J       ^.— T'^  31  1  i>  -^'      TTTl   • 

22.  What  is  the  value  of  |  of  §  1  A.  |  or  |  or  1. 

23.  Employ  both  modes  of  abbreviating  in  the  following,  viz : 
How  much  is  f  of  f  of  ^-^  of  |f  of  ^  ?  A.  yV. 

24.  A  ship's  cargo  was  valued  at  $22,000 ;  -^  of  which  in  distress 
of  weather  was  thrown  overboard ;  what  part  of  the  cargo  did  that 
man  lose  who  owned  |  of  it  ?  What  was  the  value  of  his  loss  T 
(Lii.  10.)  ^     ^.  H ;  ^^4,800. 

25.  Suppose  a  boy  can  do  a  job  of  work  in  3§  days,  and  that  a  man 

Q.  What  whole  or  mixed  numbers  are  equal  to  i^^  ?— to  Uj?  ?— to  LL?? 

Case  V,  Q.  "What  is  the  rule  for  reducing  compound  fractions  to  simple 
ones?  1.  Why  is  f  of  ^  equal  to  A|?  3.  What  is  to  be  done  when  a  whole  or 
mixed  number  occurs  ?  9.  How  much  is  ^  of  2^  ?  When  can  the  terms  be  re 
duced  or  cancelled  ?  15.    Why  ?  16.    How  much  is  -i-  of  ^  ?— of  |  ? 


REDUCTION    OF    VULGAR    FRACTIONS. 


Uo 


can  do  the  same  in  ?  of  the  time  ;  how  many  days  would  the  man  be 
in  doing  it  1  A.  Iyi. 

26.  A  father  at  his  decease  gave  f  of  his  estate,  which  was  valued 
at  ^20,000,  to  his  wife,  who  at  her  decease  gave  |  of  her  portion  to 
her  daughter.  What  part  of  the  father's  estate  did  the  daughter  re- 
ceive, and  what  was  its  value  1  A.  |  =  $7,500. 

27.  Suppose  a  man  pays  for  -^\-  of  a  ship  814,000,  and  for  f  of  its 
cargo  ^20,000,  and  subsequently  gives  f  of  all  his  interest  in  both 
ship  and  cargo  to  his  son, — 

What  is  the  son's  part  of  the  ship  and  its  value  1  A.  ^f  =$10,500. 
What,  the  son's  part  of  the  cargo  and  its  value  ]  A.  ^^  =$15,000. 
What  is  the  entire  value  of  both  ship  and  cargo  1  A.  $78, 133  j. 

CASE   VI. 

To  change  one  fraction  for  another  of  equal  value,  having  a  given 
numerator. 

RULE. 

1.  Multiply  the  numerator  of  the  required  fraction,  by  the  denom- 
inator of  the  given  fraction ;  and  divide  the  product  by  the  numerator 
of  the  sa7ne  fraction,  for.  the  required  denominator. 

2.  Reduce  f  to  an  equal  fraction  whose  numerator  shall  be  12. 

3.  If  the  I  were  \,  then  the  denom- 
1  2  given  numer.  inator  of  every  equal  fraction  would  ex-. 

" ceed  its  numerator  5  times  =  5X12  =  60, 

4  )  6  0  A.  yf.  but  I  being  4  times  as  much,  the  60  must 

1  5  required  denom.       be  decreased  on  that  account  4  times = 
'  60-^4  =  15,  the  denominator  sought. 

4.  Reduce  j^  to  an  equal  fraction  whose  numerator  is  24  A.  |^. 

5.  What  fraction,  having  20  for  its  numerator,  is  equal  to  f  l-^to 

20    20      20 
30  5  2G|'23|' 

6.  Suppose  a  company  of  105  men  purchase  |  of  a  bank,  into  how 
many  equal  shares  must  the  whole  capital  of  the  bank  be  divided  that 
each  purchaser  may  own  one  share?  A.  y|4- 

CASE   VII, 

To  change  one  fraction  for  another  of  equal  value,  having  a  given 
denominator. 

RULE. 

1.  Multiply  the  given  denominator  by  the  numerator  of  the  given 
fraction,  and  divide  the  product  by  the  denominator  of  the  same  frac- 
tion. 

2.  Reduce  §  to  an  equal  fraction  whose  denominator  shall  be  15. 

Case  vi.  Q.  What  is  the  rule  for  changing  one  fraction  for  another  with  a 
given  numerator?  1.  In  reducing  |  to  an  equal  fraction,  that  has  12  for  its 
numerator,  how  do  you  proceed  ?  3.  What  is  the  reason  for  the  operation  ?  3. 

Case  vii.     Q.  What  is  the  rule  for  finding  that  fraction  whose  denomination 
being  known  will  equal  a  given  fraction  ?  1.  Howmany  fifteenths  are  |?   Why 
do  you  multiply  15  by  4  and  divide  by  5  ?  3. 
11* 


3  1 ^to  5  ^  A. 


126 


ARITHMETIC. 


3.  If  the  i  were  j,  then  the  numerator  of  every  equal 

1  5         fraction  must  be  5  times  smaller  than  its  denominator, 

_4        that  is,  15-^5==3  ;  but  f  being  4  times  as  much  as  ^,  the 


6  )  6  0         3  must  be  increased  on  that  account  4  times  =  4  X  3  =  12, 
1  ^         the  numerator  sought ;  or  multiphj  first  by  4  and  divide 
~~      ~         hy  5  afterwards.  A.  -ff. 

4.  Reduce  |  to  a  fraction  whose  denom.  shall  be  400.      A.  fH. 

5.  How  many  sixteenths  in  £1%^  T  A.  jGyV 

6»  How  many  thirds  in  j^-  of  a  pint  1  '  zHJLs 

7.  Suppose  one  man  has  y^o^g-  of  a  barrel  of  flour ;  another  |f^ ;  a 
thirdfVVi  a  fourth  yV^V ;  afifth||f ;  and  a  sixth  ^f|:  what  frac 


CASE    VIII. 

To  Reduce  fractions  to  a  common  denominator. 

RULE. 

1.  Multiply  both  the  numerator  and  denominator  of  each  fraction 
hy  the  denominators  of  all  the  other  fractions. 

2.  For  both  terms  being  multiplied  by  the  same  numbers,  the  value 
of  every  fraction  (by  LViii.  3.)  remains  the  same;  and  the  denom- 
inator of  each  fraction  must  be  a  common  one,  for  it  is  in  each  in- 
etance,  the  product  of  the  same  numbers. 

3.  Reduce  f ,  f  and  \  to  a  common  denominatori 
fx5  denom.=^fx8  denom.  =|f^. 


^X5  denom.=^fx8  denom.  =|f^.  A. 
fx4  denom.=^x8  denom.  =^V  A. 


ix5  denom.=f|X4  denom.  =m.  A. 

4.  Since  the  reduction  of  each  fraction  involves  the  multiplication 
of  the  same  denominators,  we  need  multiply  them  together  only  once ; 
Dut  recollect  to  multiply  the  numerators  as  before)  according  to  the 
following 

GENERAL  RULE. 

5.  If  the  numbers  are  not  all  single  fractions  reduce  them  to  such 
first,  then  multiply  each  numerator  hy  all  the  denominators  except  its 
own,  for  a  new  numerator;  and  all  the  denominators  together  for  a 
new  denominator. 


6.  Reduce  f ,  f ,  and  4  to  a  com. 

2x6x7=  84.    Thenf=^. 

denominator. 

5x3x7=105.    Then^=||^. 
4x6x3=  72.     Thenf=y^^. 

A.  tVfIIM^i^V- 

3x6+7=126 

7.  Ileduce  ^,  f ,  and  f  to  a  com 

denom'r.     XyV^;  t%%\  k^. 

8.  Reduce  y^«^,  \  and  |  to  a  com 

denom'r.     ^.yVo;  ^\%\  \\l 
a  com.  denom'r.      A.  /j  ;  Vi^* 

9.  Reduce  \  off  and  f  of  11  to 

Case  viii.  Q.  Reduce  to  a  common  denominator,  |,  and  f  ;--^  and  | 
What  is  the  rule?  1.  What  is  the  illustration  of  the  rule  ?  2.  How  may  th<* 
process  be  shortened  and  why  ?  4.  What  is  the  general  rule  for  it  ?  5. 


REDUCTION    OF    VULGAR    FRACTIO.NfS.  I2t 

10.  Reduce  the  fractional  parts  of  14}  barrels,  25f  barrels,  17| 
barrels,  and  18^^j  barrels,  to  a  common  denominator. 

A       IA'150.     0^400.     17480.     IQlfitf 

^-   ^%oo'  ^^e-ocn  A/^^,j,  iw^-^Tj-. 

11.  Suppose  A  sells  -j  of  a  hogshead  of  molasses  to  B  ;  who  sells 
f  of  his  part  to  C  ;  who  sells  §  of  his  part  to  D ;  what  fractions  of  a 
hogshead  will  express  each  one's  part,  and  have  their  denominators 
all  alike  ]  A.  B  ^  ]  0  ^l ',  J)  ^%. 

CASE    IX, 
To  find  the  least  common  denominator. 
RULE. 

1.  Having  reduced  the  numbers  as  before,  find  the  least  common 
multiple  of  all  the  denominators,  for  a  common  denominator. 

2.  Then  divide  this  com.  denom.  by  the  denom.  of  each  fraction, 
and  multiply  the  quotient  by  the  numerator  for  a  new  numerator. 

3.  For  the  common  denominator  becomes,  from  being  a  product  of 
all  the  given  denominators,  a  common  multiple,  of  which  each  of  said 
denominators  is  a  factor. 

4.  Therefore  the  least  common  denominator  must  be  the  least 
common  multiple  of  the  given  denominators. 

5.  Reduce  f ,  |-,  f ,  f ,  to  their  least  common  denomitiator. 

6.  The  least  common  multiple  of  the  denominators  4,  6,  3,  and  8* 
is  24,  which  is  the  denominator  sought,  (see  lx.  40) ;  therefore, — ■ 

Com.  denom.  24-t-4x3  =  18,  new  numer.;  then  |=^|,  A. 
■6x6=20,  new  numer.;  then  f=|f,  A. 
-3X2  =  16,  new  numer.;  then  l=\xi  ^• 
■8x5  =  15,  new  numer.;  then  |— ^f,  A, 

7.  Reduce  £^,  £§,  and  ^y"^,  to  their  least  common  denominator. 

A.  £j\;  £\%;  £j\. 

8.  Reduce  $f ,  S|,  and  $||,  to  their  least  common  denominator. 

^-  $11;  $n;  m- 

9.  Reduce  |  of  f ,  f  of  8  and  4|,  to  their  least  common  denomina- 
tor "  "  42.32.27 

10.  Reduce  f ,  ^,  |,  i,  |  of  f  and  105|,  to  their  least  common  de 

4.2.3.1.3.634 


Com.  denom.  24 
Com.  denom.  24 
Com.  denom.  24 


nominator.  .  ^.1.2.3.1.3.  634 

11.  Suppose  A  owns  f  of  a  brick  block,  jd  fg-,  o  5-,  ana  u  ^; 
what  other  fractions  ha-^ing  the  least  common  denominator  will  ex- 
press each  man's  part  1  A.  ^j ;  -/g- ;  -r/^ ;  ^\ 

CASE    X. 

To  reduce  a  complex  fraction  to  a  simple  one. 

RULE. 

1.  Ifhoth  terms  be  not  single  fractions,  reduce  them  to  such  first  I 
then  to  a  common  of  least  common  denominator ;  which  strike  out 

Q.  How  do  you  proceed  with  mixed  numbers  ?  10<  What  is  a  common  de- 
nominator for  the  fractions  connected  with  6i  and  5|-? 

Case  IX.  Q.  What  is  the  rule  for  finding  the  least  common  denominator/ 
],  2.  Why  is  the  least  common  multiple  the  least  common  denominator?  3. 
What  is  the  least  common  denominator  for  g  and  4  ?— for  ?  of  |L  and  4-? 


12S  ARITHMETIC. 

^rom  both  terms,  and  the  numerators  alone  as  they  stand  will  form 
the  terms  of  the  siinple  fraction  required. 

2.  Or,  having  multiplied,  as  above,  the  numerator  of  each  term  by 
the  denominator  of  the  opposite  term,  reject  the  denominators. 

3.  For  cancelling  equal  terms  has  no  effect  on  the  value  of  the 
fraction.     See  Case  v.  15, 16. 

3 

4.  Reduce  -9-  of  a  hogshead  to  a  simple  fraction. 

5.  Reducing  the  ^  and  fy  to  a  com.  denom.  makes  ^f  and  f f ; 
then  cancelling  the  77  in  each  fraction,  we  have  33  left  for  a  nume- 
rator and  63  for  a  denominator,  all  of  which  may  be  indicated  thus : 

i-=M-=M-     Orbyrule2.    -|-=|^=M.  A.  U=h{' 

6.  Reduce  -5-  to  a  simple  fraction.  A.  ||. 

3 

7.  Reduce  -j-  to  a  simple  fraction.  A.  if =-^. 

5 

8.  Reduce  Y  to  a  simple  fraction.     (5=^.)  A.  ^=51. 

IS 

243   \  A       243 

-,-.;  A.    5^-^. 

3P6   _    99 
lT2ir— 3  3  2- 


9. 

Reduce 

341 

84 

to  a  simple  fraction. 

(34f 

10. 

Reduce 

44 

147| 

to  a  simple  fraction. 

11. 

Reduce 

247 

5 

to  a  simple  fraction. 

A.  -^-^ 


A.  I-'' 2! 


12.  Reduce  oKrifi  to  a  simple  fraction.  A.  ffff 


^09     .  .         ,       /.         ..  A       <:.80< 


13.  Reduce  -^— ^ —  to  a  simple  fraction,  (f  of  j=j%.)  JL. -gi^^i-j^. 

-  of  20^ 

14.  Reduce  I    n  .A  to  a  simple  fraction.  A^  ffll^Ufff. 

15.  When  the  numerator  is  ys  and  the  denominator  |,  what  is  the 
value  of  the  fraction,  expressed  in  its  simplest  terms  ?  A.  |-. 

16.  When  the  dividend  is  jl,  and  the  divisor  f^,  what  is  the  quo- 
tient t  A.  ||. 

17.  When  tape  is  3^2  of  a  dollar  a  yard,  how  many  yards  may  be 
bought  for  I  of  a  dollar '?  A.  28  yards. 

18.  When  f  of  8  is  a  dividend,  and  16y  a  divisor,  what  single  frac- 
tion will  express  the  quotient  1  .         ^-  ^^*^' 

19.  A  grocer  has  45|  gallons  of  cider,  which  he  wishes  to  put  into 
bottles,  each  to  hold  y*^  of  a  gallon  ;  how  many  bottles  must  he  get? 

A.   12^  dozen  bottles. 

Case  X.     Q.  What  is  the  rule  for  reducing  a  complex  fraction  to  a  simple 
one  ?  1,  2.    On  what  principle  are  the  denominators  dropped  ?  3. 


4  times  5. 

A.  h 
A. 

11. 

23   _ 
■     213- 

A 

.  1 

A. 

REDUCTION    OF    VULGAR    FRACTIONS.  129 


CASE    XI. 

To  find  what  part  one  number  is  of  another,  which  is  c^XieA  finding 
their  ratio. 

RULE. 

1.  Make  that  number  which  is  to  become  the  part,  the  numerator 
of  a  fraction,  and  the  other  number,  the  denominator,  that  is,  always 
divide  the  second  by  the  first,     liv. 

2.  What  part  of  20  is  5 1 

3.  What  part  of  5  is  20  ?  A. 

4.  What  part  of  400  is  50  1— is  200  ? 

5.  What  is  the  ratio  of  23  to  2531 

6.  What  is  the  ratio  of  253  to  23  ?  A.  -ij 

7.  What  is  the  ratio  of  832  to  624 1 

8.  What  is  the  ratio  of  624  to  832  ] 

9.  What  part  of  625  is  the  sum  of  175  and  225 1    A. 

10.  A  and  B  bought  a  barrel  of  flour  for  $11 ;  A  paid  $6  and  B 
$5 ;  what  part  of  the  whole  did  each  pay  1  A  y y  ;  B  y^y. 

11.  Suppose  A  puts  into  trade  $100,  B  $200,  and  C  $500;  what 
part  of  the  whole  capital  did  each  man  advance  \ 

A.  A|;  Bf;  C|. 

12.  Three  men  fonn  a  copartnership,  A  advancing  $1,000,  B 
$1,500,  and  C  $2,500  ;  what  is  each  one's  part  of  the  capital  1 

A.  A's^;  B'sy^'^;  C's  i. 

13.  Suppose  in  the  last  question,  that  they  gained  the  first  year 
$2,000,  which  they  are  to  share  in  proportion  to  the  sum  each 
advanced ;  how  many  dollars  did  each  gain  ?  (j  of  $2,000 =$400,  &c.) 

A.  A's$400;  B's  $600  ;  C's  $1,000. 

14.  Next  suppose  that  the  same  company  lost  the  second  year 
81,700  ;  how  many  dollars  did  each  lose  ? 

A.  A's$340;  B's  $510;  C's  $850. 

15.  A,  B  and  C  traded  together  and  gained  $3,000,  which  they 
agreed  to  share  in  proportion  to  each  one's  capital.  A  advanced 
$2,000,  B  $3,000,  and  C  $1,000  ;  how  many  dollars  did  each  man 
receive  for  his  profits  1       A.  A's  $1,000  ;  B's  $1,500  ;  C's  $500. 

16.  What  part  of  3f  gallons  is  2^  gallons  1  Reduce  the  complex 
fraction  to  a  single  one  by  Case  x.  A.  f . 

17.  A  gentleman  having  205]-  barrels  of  flour,  sold  76^f  barrels ; 
what  part  of  the  whole  did  he  sell  ?  A.  %. 

CASE    XII. 

To  find  what  fraction  or  part,  one  quantity  is  of  another  of  the  same 
kind,  but  of  different  denominations. 

RULE. 
1.  Reduce  the  given  quantities  to  the  loivest  denomination  mention- 
ed in  either  quantity ;  then  proceed  to  find  the  part  as  in  the  last  case. 

Case  XI.  Q.  What  part  of  12  is  8  ?  What  part  of  45  is  15?  What  is  the 
rule  ?  1.    What  is  the  ratio  of  20  to  1 0  ?— of  10  to  20  ?— of  40  to  120  ? 


130  ARITHMETIC, 

2.  What  part  of  ^5  is  50  cents  1    ($5  =  500ct.)     A.  $/A=iTr 

3.  What  part  of  ^1  is  2s.  6d.  1  A.  \. 

4.  What  part  of  £2.  3s.  4(1.  is  48.  2d.?  A,  /g. 

5.  Reduce  16s.  8d.  to  the  fraction  of  jCl.  A.  £|. 

6.  Reduce  18s.  to  the  fraction  of  a  guinea  (28s).  A.  yj-. 

7.  What  part  of  £1  is  Is.  Sd.l— is  2s.  6d.1— is  3s.  4d.1— is  6s.?— 
is  6s.  8d.?— is  10s.?— is  12s.  6d.?— is  15s.?  is  17s.  6d.?    , 

^-     12  J    ^»    ^>    ¥»    3^J    2  '    H  »    T»    7- 

8.  What  part  of  15  hours  is  30  minutes?  A.  ^. 

9.  What  fraction  of  3cwt.  is  3qr.?  A.  }. 

10.  What  part  of  7  miles  4  furlongs  is  Im.  2fur.?  A.  }. 

11.  What  part  of  $1  (=6s.)  is  2s.  6d.?— is  4s.  6d.?     A.  -j^;  Sf. 

12.  What  part  of  ^1  is  Id.?— is  let.?— is  Im.? 

13.  What  part  of  1  month  is  1  day  ?— is  5d.?— is  lOd.?— is  15d.? — 
is  20d.?— is  25d.?  A.  gVmo.;  |mo.;  ^mo.;  |mo.;  fmo.;  |mo. 

14.  What  part  of  2  yards  is  3qr.  3na.  A.  ff. 

15.  What  part  of  5  bushels  is  3pk.  7qt.  1  pt.?  A.  ^. 

16.  What  part  of  jC3.  15s.  6d.  is  £l.  5s.  2d.?  A.  ^. 

17.  Reduce  3qr.  181b.  12oz.  to  the  fraction  of  Icwt.  A.  ff. 

18.  What  part  of  15Y.  lOmo.  Iwk.  5d.  2h.  30m.  30sec.  is  6Y. 
4mo.  4d.  20h.  12m.  12sec.?  A.  f.     • 

19.  Reduce  7cwt.  2qr.  121b.  8oz.  to  the  fraction  of  a  ton. 

A    -4JLT 

20.  What  part  of  4d.  2fqr.  is  Id.  Ifqr.?  A.  ^. 

21.  A  merchant  bought  2cwt.  2qr.  181b.  9fdr.  of  sugar,  and  sold 
2qr.  171b.  2ffdr.;  what  part  of  the  whole  did  he  sell?  A.  j. 

CASE    XIII. 

To  reduce  the  fraction  of  any  given  quantity  to  whole  or  compound 
numbers. 

RULE. 

1.  Multiply  the  given  quantity  or  its  equivalent  in  the  next  lower 
denomination,  by  the  numerator,  and  divide  the  product  by  the  de- 
nominator;  proceeding  with  the  remainder,  if  there  be  any,  as  in 
Compound  Division. 

2.  For  dividing  by  the  denominator  first  would  show  the  value  of 
one  equal  part,  and  multiplying  the  quotient  by  the  numerator  would 
show  the  value  of  all  the  parts  meant ;  a  process  the  same  in  effect 
as  that  described  in  the  rule. 

Case  XII.  Q.  What  part  of  1  dollar  is  75  cents?— is  50  cents?— is  40  cents?— 
is  30  cents  ?— is  25  cents  ?— is  1 6|  cents  ?— is  12^  cents  ?— is  Gj-  cents  ?  What 
is  the  rule  ?  1.  What  part  of  £1  is  5s.?— is  2s.  6(1.?— is  6s.  8d.?— is  13s.  4d.?— 
is  15s.? 

Case  XIII.  Q.  How  many  cents  are  there  in  |  of  a  dollar  ?— shillings  m  | 
of  £1.  What  is  the  rule?  1.  Why  multiply  by  the  numerator  and  divide  by 
the  denominator?  2.  How  many  seconds  are  there  in  |^  of  a  minute  ? — fur- 
longs in  ^  of  a  mile  ?— feet  in  |  of  a  yard  ? 


REDUCTION  OF  VULGAR  FRACTIONS.         131 

3.  How  much  is  f  of  a  day  1     1  day =24  hours  ;  then  24  hours  X 
3-^4=^18.  A.   18  hours. 

4.  What  is  the  value  of  f  of  a  shilling  ■?  A.  9  pence. 

5.  What  is  the  value  of  f  of  jCl  ]  A.   12  shillings. 

6.  What  is  the  value  of  f  of  £1?  A.  13s.  4d. 

7.  Suppose  a  railroad  car  goes  10  miles  in  §  of  an  hour,  how  many 
*minutes  is  it  in  going  that  distance  1  A.  2i  minutes. 

8.  What  is  the  value  of  t^V  of  5  dollars  1  A.  20  cents. 

9.  How  much  is  jCj^i— ^1  t.—£^g—£l  1— jG-J-  ^.—£1  ? 

Answers:   Is.  8d.;  2s.  6d.;  3s.  4d.;  5s.;  6s.  8d.;  17s.  6d. 

10.  How  many  shillings  and  how  many  cents  are  equal  to  -j^  of  a 
doUarl  ^.  2s.  6d.=41f  cents. 

11.  How  many  days  are  equal  to  ^  of  a  month? — to  |  of  a  month  1 
to  f  of  a  month  1  ^.  6d.;  25d.;  22^. 

12.  How  much  is  |  of  7  miles  4  furlongs  ?  A.  Im.  2fur. 

13.  How  much  is  ^|  of  2  yards  ?  A.  3qr.  3na. 

14.  How  much  is  -^^  of  5  bushels  1  A.  3pk.  7qt.  Ipt. 

15.  Suppose  a  grocer  buys  |  of  a  hogshead  of  molasses ;  how 
many  gallons,  quarts,  &c.  does  he  buy'?  A.  39gal.  Iqt.  Ipt. 

16.  What  is  the  value  of -f  of  a  pound  avoirdupois  1 

A.  lloz.efdr. 

17.  What  is  the  value  of  yj  of  a  day?      A.  16h.  36m.  55y^3sec. 

18.  Suppose  you  resided  in  one  place  4Y.  3mo.  3d.,  and  in  another 
place  Y  as  long ;  what  period  of  time  did  you  spend  in  the  latter  place  1 

A.  7mo.  9d. 

19.  What  is  the  value  of  ?  of  2hhd.  27gal.  2qt.  Ipt.  3gi.? 

A.  43gal.  3qt.  Ipt.  Ifgi. 

CASE   XIV. 

To  reduce  a  fraction  of  one  denomination  to  an  equivalent  fraction 
of  another  denomination. 

RULE. 

1.  Multiply  and  divide  the  fraction  according  to  the  principles  of 
Reduction  of  who^numbers,  but  recollect — 

2.  That  a  fraction  is  multiplied  by  multiplying  its  numerator,  or 
by  dividing  its  denominator,     lvii.  17. 

3.  Also,  that  a  fraction  is  divided  by  dividing  its  numerator,  or  by 
multiplying  its  denominator,     lvii.  18. 

4.  Reduce  j-^t^  of  a  pound  to  the  fraction  of  a  farthing. 
_^  1    X20s.xl2d.x4qr.      960 

Thus:  ^1920 ""1920^^- '"^^^-  ^'  ^^' 


Or  thus:  ^i920-^20s.-^12d-^4qr.^^^- '^'  ^^^^ 

Case  XIV.  Q.  What  part  of  £1  is  1  of  a  shilling?  What  part  of  a  gallon 
is  ^  of  a  quart?  What  is  the  rule?  f.  How  is  the  fraction  multiplied  or 
divided?  2,  3.     How  is  y^U^  of  a  £  reduced  to  the  fraction  of  a  farthing?  4. 


132  ARITHMETIC. 

5.  Reduce  ^%%  (=1)  of  a  farthing  to  the  fraction  of  a  pound. 

960^4qr.^l2d.-20s. 
^'^"'=   1920 =-^Ti>W-  A.  £j^j,. 

^     ,  960  

Or  t  nil's  • 000      — -pi  A     -p    i 

"lub.   i920x4qr.Xl2d.-^20s.~T?^i^^^'"^To^«"-      ^-  ^T?^ 

060      1 

Urthiis:   1920 ~2x4qr.xl2d.x 20s. ~"^T^^  ^-  ^n^ 

6.  Reduce  g^o  of  a  pound  to  the  fraction  of  a  shilling. 

7.  Reduce  -jV  of  a  shilling  to  the  fraction  of  a  pound. 

8.  Reduce  j-^^jfo  of  an  eagle  to  the  fraction  of  a  mill. 

9.  Reduce  f  of  a  mill  to  the  fraction  of  an  eagle. 

10.  Reduce  yo  fxa  of  a  guinea  to  the  fraction  of  a  farthing. 
H.  Re(hice  |  of  a  farthing  to  the  fraction  of  a  guinea. 

12.  Reduce  ^  of  a  pound  to  the  fraction  of  a  guinea. 

13.  Reduce  ^  of  a  guinea  to  the  fraction  of  a  pound. 

14.  Reduce  yyVy  of  a  day  to  the  fraction  of  a  minute. 

15.  Reduce  {^1^-  of  a  minute  to  the  fraction  of  a  day. 

16.  Reduce  yfy  of  a  bushel  to  the  fraction  of  a  quart. 

17.  Reduce  Iff  of  a  quart  to  the  fraction  of  a  bushel. 

18.  Suppose  one  man  has  jl^  of  a  pound,  another  oV  of  a  shilling, 
and  another  4^  of  a  penny ;  what  fraction  of  a  dollar  Has  each. 

^  "  A,    $yl,. 

19.  Reduce  ^  of  a  pound  to  the  fraction  of  a  crown  at  6s.  8d.  each. 

A.  11=1  crown. 

20.  What  fraction  of  a  pound  is  ^o  of  I  of  a  hundred  weight  1 

A.  -5^1b.  =3  pounds. 

21.  Suppose  you  owe  f  of  a  guinea,  (28s.)  and  pay  ^  of  the  debt ; 
what  part  of  a  pound  do  you  still  owe !  A.  £j^. 

22.  If  A  buys  j\  of  5  hogsheads  of  molasses,  and  sells  B.  ^^  of  it, 
what  fraction  of  a  gallon  does  B  buy?  A.  //g-  of  a  gallon. 

23.  Suppose  f  of  |  of  a  pound  isthe  numerator  of  a  fraction,  and 
Yff  of  £2\  the  denominator  ;  what  fraction  of  an  eagle  will  express 
the  same  value  1  A.  ff  of  an  eagle. 

CASE   XV.  • 

To  find  the  integer  from  having  a  fractional  part  given. 

RULE. 

1.  Multiphj  the  integer  hy  the  denominator^  and  divide  the  product 
by  the  numerator,     lii.  10. 

2.  Find  that  number  Y^3  of  which  is  205.  A.  888^. 

3.  What  number  is  that  -^  of  which  is  20,000 1         A.  55,000. 

4.  Suppose  a  merchant  sells  exactly  llhhd.  49gal.  3qt.  Ipt.  3gi. 
of  molasses,  it  being  ^\  of  all  he  has  on  hand ;  what  quantity  had  he 
at  first  ? "_ A.  78hhd.  39gal.  Ipt. 

Q.  How  is  ■^■^~  (  =^)  of  a  farthing  reduced  to  the  fraction  of  a  pound?  8, 9. 

Case  XV.  QT.  40  is  i  of  what  number?  If  12  be  ^  of  a  certain  number, 
what  is  that  number?  What  is  the  rule?  1.  When  a  man  pays  8  dollars  for  | 
of  a  barrel  of  flour,  what  is  a  barrel  worth  at  that  rate  ? 


ADDITION    or    FRACTIONS.  133 

.   5.  Find  that  sum  |  of  which  is  £3.  5s.  6|d.  A.  £5.  17s.  Hid. 

6.  The  postage  of  a  letter  for  400  miles  is  ^  of  a  dollar  ;  how  far, 
at  that  rate,  can  a  letter  be  carried  for  one  dollar]  A.  2, 133 j miles. 

7.  503  is  ^  of  y\  of  what  number?  A.  1475yV 

8.  It  is  estimated  that  nearly  25,000,000  persons  die  annually, 
being  3V  of  the  population  of  the  earth ;  what,  according  to  that  esti- 
mate, must  be  the  whole  population  of  the  globe  1  A.  800,000,000. 


ADDITION   OF   FRACTIONS. 

GENERAL    RULE. 

LXIII.  1.  Reduce  complex  and  compound  fractions  to  single 
ones,  and  all  to  a  common  or  least  common  denominator ;  over  which 
write  the  sum  of  the  numerators,     lv.  1,2. 

2.  This  sum  may  often  be  reduced  either  to  lower  terms,  or  to  a 
whole  or  mixed  number. 

3.  Add  together  £§ ,  £^  jCf  and  £\.  A.  £  V  =£'2. 

4.  Add  together  ^||,  ||f,  A^,  and  l\\.  A.   l^f 

5.  To  add  mixed  numbers. — Find  the  sum  of  the  fractional  parts 
first,  and  if  it  contain  a  tvhole  number,  add  it  to  the  sum  of  the  other 
whole  numbers. 

6.  Or,  reduce  the  mixed  numbers  first  to  improper  fractions,  then 
add  them  by  the  general  rule. 

T.  Add  together  $1,203||,  S9,406|-^,  and  $8,906||. 

S  1  2  0  3  —'h 

$  0  4  0  6  ||  8.  The  fractions  make  ${f =$2/-ff=S2|; 

$  8  9  0  6  If        carry  the  12  to  the  column  of  dollars. 
S  1  9  5  1  7  i  A.  19,517f 


9.  Find  the  sum  of  675|i,  89 If 4,  and  915f|.  A.  2,483^- 

10.  Suppose  that  one  pile  of  wood  contains  136H  cords,  another 
956  If  cords,  and  another  45211-  cords  ;  how  many  cords  are  there 
in  all  the  piles'?    •  A.  l,545f|  cords. 

11.  Add  together  f,  f,  ^,  and  f.  Reduce  them  first  to  their  least 
common  denominator.  A.  2}^.     - 

12.  Find  the  sum  of  46f ,  89f  and  40  1.  A.  1775-V 

13.  Add  together  $18f,  $5^  $7y\,  S8f,  and  $9^.       A.  $505-^. 

14.  What  is  the  sura  of  789  f  years,  817f  years,  316|  years,  and 
S16^  years  1  A.  2,140^. 

15.  Find  the  sum  of  f ,  f  of  f ,  and  |  of  5|.  Reduce  the  fractions 
to  single  ones  first.  A.  4}^f. 

LXIII.  Q.  What  is  the  general  rule  for  adding  fractions?  1.  What  may 
oftentimes  be  done  with  the  result  ?  2.  What  is  the  sum  of  ^,  -j^-,  -i,  and  -j-^? 
What  is  the  sum  of  84,  9|,  and  10  j  ?  What  is  the  rule  for  the  last  example  ?  5, 6. 
What  is  the  sum  of  4-  and  #?  See  11. 


12 


134  ARITHMETIC. 

16.  What  is  the  sura  of  819?  barrels,  ^  of  I  barrels,  409|  barrels, 
and  \  of  5/  barrels ]  A-  1,232  barrels. 

17.  Add  together  -5-*  -3-'  ^"ci  X"     Reduce  them  first  to  single 

fractions,  then  proceed  as  before.  A.  15jf . 

18.  Add  together  the  complex  fractions  that  may  be  formed  by 
dividing  2  by  1,  i  by  4,  f  by  |,  and  2^  by  3.  A.  5j\\. 

19.  To  add  fractions  of  different  denominations. — First  find  the 
value  of  each  in  compound  numbers^  then  add  them  as  in  Compound 
Addition. 

20.  Or,  first  reduce  the  fractions  of  different  denominations  to 
those  of  the  same,  then  add  them  by  the  general  rule. 

21.  Add  together  I  of  a  pound  and  f  of  a  shilling.  jC^=3s.  4d. — 
fs.=8d.     Then3s.  4d.  +  8d.=4s.  A.  As. 

22.  Add  together  £^-  and  /f  of  a  shilling.         A.  8s.  lOd.  l2qr. 

23.  Find  the  sum  of  yy  of  a  ton,  y  of  a  hundred  weight,  and  |  of  a 
pound.  A.  llcwt.  2qr.  131b.  3oz.  6f4dr. 

24.  Find  the  sum  of  f  of  a  league,  yy  of  a  yard,  and  f  of  a  foot. 

A.  Im.  6fur.  16rd.  2ft.  lin.  iJj-b.c. 

25.  Add  together  §  of  an  ounce  and  f  of  an  ounce.     A.  14/^dr. 

26.  A  man  labored  10|  hours  in  one  day,  9yy  hours  in  another, 
and  llf  hours  in  another ;  how  many  hours  did  he  labor  in  all  ? 

A.  31h.  35m.  13^fsec. 


SUBTRACTION    OF    FRACTIONS. 

GENERAL    RULE. 

LXIV.  1.  Reduce  complex  and  compound  fractions  to  single 
ones,  and  all  to  a  common  or  least  common  denominator,  over  lohich 
write  the  difference  of  the  numerators,     lv.  1,  2. 

2.  From  y^^  of  a  dollar  take  y^^  of  a  dollar.  A.  $i%=l. 

3.  From  UU  take  Hf |.  a.  Iff! . 

4.  From  ^^  take  f |f|.  A.  ffff. 

5.  From  ^  take  gy.  Reduce  them  first  to  their  least  common  de- 
nominator. A.  If. 

6.  From  |  take  ^x-  A.  ^. 

7.  From  4  take  ■^.  A.  \^. 

8.  A  bought  y^  of  a  load  of  hay,  and  B.  |  of  a  load ;  how  much  did 
one  buy  more  than  the  other  1  A.  A.^^  the  most. 

Q.  How  are  fractions  of  different  denominations  added?  19,  20.  What  is 
the  sum  of  £|  and  3  of  a  shilling  ?  What  is  the  sum  of  |  of  a  mile  and  ||  of 
a  furlong? 

LXIV.  Q.  From  f  take  2.  From  ^  take  ^.  What  is  the  rule?  1.  From 
I  of  I  of  a  dollar  take  yL  of  a  dollar. 


SUBTRACTION    OF    FRACTIONS.  135 

9.  From  f  of  5|  take  I  A.  ^1=2^1 

10.  From  4~  take  |  of  3.  "  A.  2}. 

11.  A  having  bought  a  quantity  of  sugar,  sold  ^  of  it  to  B,  who 
sold  f  of  what  he  bought  to  C  ;  what  part  had  B  left  ?  A.  y\. 

2  g  3. 

12.  From  -f-  take  ^-     From  -^  take  -^• 

w  f  ^ 

13.  To  subtract  a  mixed  number  from  a  whole  number. — Write 
the  difference  between  the  numerator  and  denominator  over  the  de- 
nominator, and  carry  1  to  the  whole  number. 

14.  For,  since  the  denominator  expresses  all  the  parts  of  1  unit, 
the  process  is  the  same  in  pinciple,  as  borrowing  and  carrying  in 
simple  numbers. 

15.  From  $  8  1  5  ^^y,  5  from  16^11,  that  is,  ||  and 
Take  $ 3^6^         carry   1.     For  W  from  -f^  (^1  unit 

A.  %  1  1  ^  -Ti         borrowed  from  815)  leaves  j^  and  1  to 
j^         carry  as  before. 

16.  From  £5,075  take  i:3,536}f.  A.  jC2,538ff. 

17.  From  487  years  take  259^  years.  A.  221^^^  years. 

18.  From  1  league  take  |f  of  a  league.  A.  \j  league. 

19.  How  much  less  than  unity  is  f^^f  1  A.  ^oVo- 

20.  From  1  take  the  sum  of  |  and  |.  A.  f . 


21.  From  1  take  the  sum  off  and  ■^^.  A.  -jV 

22.  What  fraction  added  to  the  sum  of  f  and  g^j  will  make  a  un: 
^  A.  \i. 


y  23.  Suppose  a  pole  standing  so  that  \  of  it  is  in  the  mud,  f  in  the 
/water,  and  the  rest  above  water  ;  what  part  is  above  water  1  A.  \'J . 
I         24.  From  1  take  the  sum  of  f\  and  f,  A.  /t^. 

/  25.  A  gentleman  spent  \  of  his  life  at  school,  \  in  England,  \  in 

i       America,  and  the  rest  in  France.     What  part  of  his  life  did  he  spend 
^■— -in  the  latter  country  \  A.  \%. 

26.  To  subtract  one  mixed  number  from  another. — Having  re- 
duced the  fractions  to  a  common  denominator,  subtract  as  at  firsts 
hut  if  the  lower  numerator  exceed  the  upper  one,  subtract  it  from  the 
common  denominator  and  add  the  difference  to  the  upper  numerator, 
carrying  one  as  before. 

27.  From  JC  9  6  |^  Say,  5  from  11=6,  that  is,  .'V,  or  ^^ 
Take  jg  1  8  tV        hom\\=fs  =  l' 

£78  A =785,  Answer. 


28.  From  608yVV  take  504f.%.  A.  104yV\=^1042V 

29.  A  merchant  owing  $405^,  paid  S293| ;  how  many  dollars  re- 
main unpaid ? A.  \\2\. 

Q.  How  is  a  mixed  nuinbrr  s'abtr.irtp*!  from  a  whole  numVior  ?  13.  Why  is 
the  numerator  to  be  svibtractrd  from  the  denominator?  14.  From  4  gallons  take 
2-py  gallons.  From  £1  take  £^^,-.  How  much  does  unity  exceed  ttjtI 
How  is  one  mixed  number  sulttracted  from  another?  26.  A  man  having  4| 
yards  of  broadcloth,  sold  21  yards  ;  how  many  yards  had  he  left?  From  4^ 
take  ]| 


136  ARITHMETIC. 


30.  From  jC  1  5  0  5- 

T 


Take  jC  *  7  5  |  Say,  6  from 7  =  1+2=3, that  is, f,orf 

Ans.  £      7~T~4        from"^=|+f =f.     Carry  1  to  the  5. 

31.  From  ISOyVr  take  107yW  A.  42||f. 

32.  From  4|  take  S^-  Reduce  the  fractional  parts  to  the  least 
common  denominator  first.  A.  -j4. 

33.  From  the  sum  of  $45§  and  S62f  take  $49^.  -      A.  $59^^^. 

34.  A  merchant  purchased  21  ^  barrels  of  flour  of  one  man,  and 
13y\  barrels  of  another ;  how  much  will  he  have  left  after  he  has 
sold  30|  barrels  ?  A.  ^  barrels. 

35.  To  subtract  fractions  of  different  integers. — First  find  thei? 
separate  values,  then  subtract  as  in  compound  numbers. 

36.  From  j  of  a  day  take  |  of  an  hour.  A.   18h.  27m. 

37.  From  £^  take  f  of  a  shilling.  A.  8s.  4d.  If  qr. 

38.  When  a  man  has  traveled  4  of  the  distance  from  New  York  to 
Philadelphia,  it  being  90  miles,  how  far  has  he  still  to  travel  1 

A.  38m.  4fur.  22rd.  4yd.  2ft.  lin.  2]h.c. 


MULTIPLICATION    OF    FRACTIONS. 

LXY.  1.  To  multiply  a  fraction  by  a  whole  number. — Multiply 
the  numerator,  or  rather  divide  the  denominator  when  it  can  be  done 
without  a  remainder,     lvii.  17. 

2.  If  you  pay  ^  of  a  dollar  for  a  yard  of  ribbon,  what  will  2  yards 
cost  1—4  yards  cost  1—8  yards  cost?  A.  $1;  $| ;  $1. 

3.  Multiply  ^0  by  10  ;— by  4  ;— by  8.  A.  ^^ ;  ^f /;  yfrr- 

4.  Multiply  -pf^  by  3  ;-by  9  ;-by  7.  A.  jU;  tVt  ;  -AS- 

5.  All  fractional  answers  should  be  expressed  in  their  lowest  terms. 

6.  Multiply  ^^  by  4  ;— by  12  ;— by  14.  A.  ^\\ ;  i^j ;  tVV 

7.  Multiply  f  by  8  ;— by  6  ;— by  4.  A.  S  ;  2^  ;   l|. 

8.  If  one  horse  consume  ^^i  of  a  ton  of  hay  in  a  month,  how  much 
'will  7  horses  consume  in  the  same  time  ■?  A.   1^  ton. 

9.  To  multiply  a  whole  number  by  a  fraction. — Multiply  by  the 
numerator  and  divide  by  the  denominator,  or  divide  first  when  it  can 
be  done  without  a  remainder. 

10.  This  rule  proceeds  on  the  same  general  principle  as  that  for 
whole  numbers,  viz. 

11.  That  as  many  times  as  the  multiplier  is  made  smaller,  so  many 
times  the  product  is  ?nade  smaller. ^^ ^^ 

Q.  How  are  fractions  of  diflferent  integers  subtracted  ?  35.  From  i  of  a  day 
take  J  of  a  minute.  From  4  dollars  take  '^  of  a  dollar.  From  1  of  I  of  Icwt 
take  3  of  ]  qr. 

LXV.  Q.  How  is  a  fraction  multiplied  by  a  whole  number?  1.  Multiply 
^^  by  5  ; — /,-  by  5  ; — g^  by  320.  How  is  a  whole  number  multiplied  by  a 
fraction?  9."  On  what  principle  is  the  rule  based  ?  11. 


MULTIPLICATION    OF    FRACTIONS.  137 

12.  When  the  multiplier  is  1,  for  instance,  the  product  is  the  same 
as  the  multiplicand. 

13.  But  in  multiplymg  by  |,  for  example,  the  multiplier  is  8  times 
less  than  unity  ;  consequently  the  product  must  be  8  times  less  than 
the  multiplicand. 

14.  In  multiplying  by  i,  the  multiplier  is  3  times  greater  than  | ; 
consequently  the  product  must  be  made  3  times  greater  than  that  of  |. 

15.  Multiply  64  by  1  ;— by  1 ;— by  i  ;— by  | ;— by  j\  ;— by  ^  ;— 
by  A ;— and  by  /^ ;— by  A  ;-by  A  ;— by  If ;— by  ^f  ;-by  ||. 

16.  Answers.    64;  32;  16;  8;  4;  2;   1;  2;  4;  8;  16  ;32;  64. 

17.  If  a  laborer  receive  20  dollars  a  month,  and  saves  fV  ^^  i^?  how 
many  dollars  does  he  save  ?  A.  $6. 

18.  Multiply  43  by  i  ;— by  I ;— by  |.  A.   16^;  34f;  28f. 

19.  Multiply  101  by  ,\  ;— 319  by  |.  A.  Q^^;   141^. 

20.  Since  either  factor  in  multiplication  may  be  made  the  multiplier 
without  affecting  the  result,  therefore, 

21.  If  more  convenient,  multiply  the  fraction  hy  the  whole  number 
instead  of  the  whole  number  by  the  fraction. 

22.  How  much  is  y%  of  6  ?— 6  times  y%  ]  A.  ^  each. 

23.  How  much  is  f  ^-|  of  180,  or  180  times  f|4  ^  A^  100^. 

24.  To  multiply  one  fraction  by  another. — Multiply  the  numerators 
together  for  a  new  numerator,  and  the  denominators  together  for  a 
new  denominator,     lxii.  case  v.  1. 

25.  Multiply  11^  by  llf  •  A.  y^A- 

26.  Multiply  l-^ll  by  II  A.  m^. 

27.  How  much  is  f  of  400  ?  A.  240. 

28.  How  much  is  j  of  208]  A.   166f. 

29.  How  much  is  f  of  5}  ]  A.  4|. 

30.  Multiply  f,  I,  f  and  |  together.       A.  ^.  lxii. — case  v.  15. 

31.  A  merchant  had  f  of  a  yard  of  cloth  in  one  remnant,  and  in 
another  only  f  as  much  ;  what  part  of  a  yard  was  there  in  the  smaller 
piece  ]  tI.  1^  of  a  yard. 

32.  To  multiply  a  mixed  number  by  a  whole  number,  and  vice 
versa. — Multiply  the  ivhole  number  first  by  the  fraction  of  the  mixed 
number,  then  by  the  whole  number  connected  with  the  fraction,  and 
add  the  two  products  together. 

8 

9  f  33.  Multiply  8  by  9|,  or  9f  by  8.     First  find  f 

5^  of8=5i,  &c.  Or,9f  =  2_«x8=^§-=77],Answer: 

7  2  that  is,  reduce  first  to  an  improper  fraction,  then 

Ans.  7  7  i  multiply  as  before. 


Q.  Why  should  any  product  be  less  than  the  multiplicand?  13.  Why  is  the 
product  of  I  greater  than  \  1  14.  What  is  the  product  of  12  multiplied  by  |? — 
by  I?— by  I?  Why  is  ^  of  6  the  same  as  6  times  -^^1  20,  21.  How  is  one 
fraction  multiplied  by  another  ?  24.  Multiply  |  by  | ;— "J  by  | ;— 1  of  |  by  j  ;— 
by  5i ;— by  f.  How  much  is  9|  times  8  ?  What  is  the  rule  ?  32. 
12* 


138  ARITHMETIC. 

34.  Multiply  92  yards  by  27f .  A.  2,535j-  yards. 

35.  Multiply  8^cwt.  by  5.  A.  43fcwt. 

36.  Suppose  45  hogsheads  of  molasses  contain  each  37|  gallons ; 
how  many  gallons  do  all  the  hogsheads  contain  ?        A.   l,68l|gal. 

37.  Reduce  241  rods  to  yards.  A.   1,325|^  yards. 

38.  Reduce  65  barrels  to  gallons.  A.  2,047|  gallons. 

39.  Reduce  40  sq.  rods  to  sq.  feet.  A.   10,890  sq.  feet. 

40.  Reduce  34x7  n^iles  to  furlongs.  A.  273^^  furlongs. 

41.  Reduce  45,  minutes  to  seconds.  A.  2,725,-  seconds. 

42.  In  some  cases  it  is  more  convenient  to  multiply  the  fraction 
first  by  the  whole  number  as  above.  (1.) 

43.  Multiply  4  0  8  3  2  | 

by  9  Thus:  |x  9  =  ^  =  ^1;  carry  the  5. 

Answer,  3  6  7  4  9  3  | 


44.  Multiply  2,51 7,937|if  by  98.  A.  246,757,882|^. 

45.  Multiply  8,300,675^^2'^  by  22.  A.  182,614,850fl. 

46.  When  both  terms  are  mixed  numbers. — Reduce  them  to  im- 
proper fractions,  then  multiply  as  above.  (24.) 

47.  How  many  yards  are  there  in  5^  rods  1  A.  3 If  yards. 

48.  How  many  square  yards  are  there  in  a  carpet  which  is  b\ 
yards  square  ]  A.  30^  sq.  yards. 

49.  How  many  square  rods  in  a  piece  of  land  which  has  four  sides, 
each  25f  rods  long '?  A.  643|J. 

50.  How  many  square  feet  are  there  in  a  room  16i  feet  square? 

A.  272-1-  sq.  feet. 

51.  How  many  solid  feet  are  there  in  a  block,  the  length,  breadth 
and  depth  of  which  are  each  3|  feet  T  A.  492^  solid  feet. 

52.  To  multiply  complex  fractions. — Reduce  them  first  to  single 
terms,  then  proceed  as  above  directed.  (24.) 

6 

63.  Multiply  ^  by  100.  A.  6^. 

8 

54.  Multiply -^  by  200.  A.  2,400. 

55.  Multiply  1  of  -^  by  3.  A.  \\. 

56.  Suppose  A,  who  owns  f  of  a  ship,  divides  his  interest  into  10 
equal  parts ;  what  fraction  of  the  whole  ship  will  express  f  of  each 
of  these  equal  parts  T  A.  jV- 

Q.  What  is  the  cost  of  4|  yards  of  broadcloth  at  $10  a  yard?  How  can  this 
process  be  varied  ?  42.  Multiply  in  this  manner  2|  by  5.  How  are  complex 
fractions  multiplied  ?  52. 


DIVISION    OF    FRACTIONS.  139 


DIVISION    OF    FRACTIONS. 

LXVI.  1.  Division  of  Fractions,  like  that  of  whole  numbers, 
proceeds  on  the  same  general  principle,  viz. 

2.  That  as  many  times  as  the  divisor  is  made  smaller^  so  many 
times  the  quotient  is  made  greater,  and  the  reverse. 

3.  When  the  divisor  is  unity,  the  quotient  is  of  course  the  same  as 
the  dividend. 

4.  But  v^^hen  the  divisor  is  any  number  of  times  less  than  unity, 
the  quotient  is  that  number  of  times  greater  than  the  dividend. 

5.  For  instance,  there  are  four  times  as  many  quarters  of  an  apple 
in  a  basket  as  there  are  whole  apples. 

6.  When  the  divisor  and  dividend  are  alike,  the  quotient  is  of 
course  unity. 

7.  But  when  the  divisor  is  any  number  of  times  greater  than  the 
dividend,  the  quotient  is  that  number  of  times  less  than  unity. 

8.  For  though  the  dividend  cannot,  strictly  speaking,  be  said  to 
contain  the  divisor,  it  may  nevertheless  be  divided  into  any  assign- 
able number  of  equal  parts,  as, — 

9.  For  example,  1  cannot  contain  either  2,  3,  or  4,  but  it  may  be 
divided  into  either  2,  3,  or  4  equal  parts,  as  i,  ^,  |. 

10.  To  divide  a  fraction  by  a  whole  number. — Divide  the  nume^ 
rator  or  multiply  the  denominator,     lvii.  17,  18. 

1 1 .  Recollect  to  reduce,  in  all  instances,  complex  and  compound 
fractions  to  single  ones,  and  mixed  numbers  to  improper  fractions, 
unless  otherwise  directed. 

12.  When  3  bushels  of  oats  cost  ^  of  a  dollar,  how  many  eighths 
will  buy  one  bushel '?  A.  $\. 

13.  Divide|by5;— by7;— byll.  A.  i;  ^;  ^V 

14.  Divide  tV/t  by  27,  and  iff  by  45.  A.  j^j ;  ^^. 

15.  Divide  §  of  5h  by  3  ;— by  33.  A.  lA ;  A- 

IG.  Divide  -J  by  6,  and  -f  by  10.  A.  ^V  ;  A- 

17.  A  man  having  a  tierce  of  molasses  containing  42^  gallons, 
sold  ^  of  it  to  a  grocer,  who  retailed  it  out  in  equal  portions  to  20 
diiferent  persons.  What  part  of  the  hogshead  did  each  of  the  20 
persons  buy]  A.  -jVw^hd. 

18.  To  divide  a  whole  number  by  a  fraction. — Multiply  by  the 
fraction  inverted,  that  is,  multiply  by  the  denominator  and  divide  the 
result  by  the  numerator. 

LXVI.  Q.  On  what  principle  is  Division  of  Fractions  based?  2.  Why 
should  the  quotient  ever  be  greater  than  the  dividend?  3, 4.  Give  an  example.  5 
When  is  the  quotient  less  than  unity,  and  why  ?  6,  7.  But  can  the  greater 
contain  the  less  ?  8.  Give  an  example.  9.  How  is  a  fraction  divided  by  a 
whole  number?  10.  What  preparation  is  often  necessary?  11.  Divide  J^|  by 
8 ; — by  4 ; — by  3 ;— by  10.  Divide  |  of  j^|  by  5 ; — by  1 5.  How  is  a  whole  number 
divided  by  a  fraction?  18. 


140  ARITHMETIC. 

19.  For  instance,  when  we  divide  by  |  the  divisor  is  8  times  less 
than  unity ;  consequently,  the  quotient  must  be  8  times  greater  than 
the  dividend.  (4.) 

20.  But  when  we  divide  by  |,  the  divisor  is  3  times  greater  than  | ; 
consequently,  the  quotient  must  be  made  3  times  smaller.  (1,  2.) 

21.  At  y  of  a  dollar  a  yard,  how  many  yards  of  calico  may  be 
bought  for  $1 1— for  $5  ]— for  $17 1— for  $4001 

A.  5yd.;  25yd.;  85yd.;  2,000yd. 
,     22.  Divide  563,015  yards  into  pieces  containing  only  ^  of  a  yard 
/each.  A.  2,412,921 1  pieces. 

23.  Divide  908,070  by  |i.  A.  930,491^^. 

24.  Divide  8,450  by  i  of  3|.  A.  12,071f. 

25.  Divide  8,307  by  ^-  A.  12,460^. 

26.  How  many  times  does  85  exceed  f?  A.  113^  times. 

27.  Divide  100  by  2;— by  1;— by^;— by  ^  ;— by  | ;— by  ^V  ;— 
by  ^.  A.  50;  100;  200;  400;  800;  1600;  3200. 

28.  Reduce  221  yards  to  rods.     (5i=y.)  A.  40^2^  rods. 

29.  Reduce  13,945  degrees  to  statute  miles.  A.  200y^5"^. 

30.  Reduce  69,563  sq.  yards  to  rods.  If  we  divide  the  remainder 
by  the  denominator,  (which  is  the  multiplier  of  the  dividend,)  the 
quotient  will  be  of  the  same  denomination  with  the  dividend. 

A.  2,299  sq.  rd.  18^  sq.  yd. 

31.  Reduce  1,229  sq.  yards  to  sq.  rods.    A.  40  sq.  rd.  19  sq.  yd. 

32.  Reduce  30,052  gallons  to  barrels.  A.  954  bl.  1  gal. 

33.  How  many  years,  of  365|  days  each,  are  there  in  28,567  days  1 

A.  78  Y.  77-^d. 

34.  If  one  bushel  of  apples  will  make  y\  of  a  barrel  of  cider,  how 
many  bushels  will  be  required  to  make  250  barrels  ?      A.  l,416f. 

35.  To  divide  one  fraction  by  another. — Invert  the  divisor,  then 
multiply  the  upper  terms  together  for  a  new  numerator,  and  the  lower 
terms  together  for  a  new  denominator. 

36.  The  reason  for  this  rule  is  the  same,  in  reality,  as  that  for  the 
preceding  one. 

37.  For,  multiplying  the  numerator  of  the  dividend  by  the  denomi- 
nator of  the  divisor  multiplies  the  dividend  by  that  number,  (lvi.  2,  3.) 

38.  And  multiplying  the  denominator  of  the  dividend  by  the  nume- 
rator of  the  divisor  divides  the  dividend  by  that  number,  (lvii.  2,  3.) 

39.  When  a  yard  of  ribbon  costs  -j^  of  a  dollar,  how  many  yards 
will  I  of  a  dollar  purchase  ? 

40.  The  divisor  ,3^  inverted=V^|=f^=4yV  A.  4^  yd. 

41.  Divide  I  by  f^^  ;— f  by  |. A.  2^\;  f. 

Q.  Illustrate  the  rule  by  the  divisors  |  and  |.  19,  20.  Divide  40  by  | ;— by  |. 
When  wood  is  6  dollars  a  cord,  what  is  1  of  a  cord  worth  ?  How  is  one  fraction 
divided  by  another?  35.  What  is  the  proof  that  this  rule  is  the  same  in  prin- 
ciple as  the  last?  37,  38.  Divide  f  ^Y  ^  ; — -^  by  |.  How  may  this  process 
be  abbreviated?  50, 


DIVISION    OF    FRACTIONS.  141 

42.  Divide  I  by  f ;— f  by  f .  A.  H ;  |f . 

43.  Divide  5|  yards  into  pieces  containing  each  ^  of  a  yard. 

A.  12f  pieces. 

44.  If  a  barrel  of  molasses  leak  out  f  of  a  gallon  a  week,  how 
many  weeks,  at  that  rate,  must  elapse  before  the  whole  will  have 
leaked  out  1  ^.  47^  weeks. 

45.  Divide  30}  by  5^  and  272^  by  16^.  A.  5^ ;  16f 

46.  How  many  bottles,  each  holding  2^  pints,  may  be  filled  with 
503 1  pints?  A.  239^. 

47.  How  many  measures,  each  holding  2f  pints  of  grain,  may  be 
filled  with  5  bushels  3  pecks  and  if  pints  of  grain.  A.  1363^. 

48.  How  many  paces,  each  equal  to  ^  of  7Yy  feet,  are  there  in  1 
mile]  1, 489 Y^  paces. 

-r..  .,      12    ,       6       ^.  12  X  7       84       2 

49.  Divide  -35-  by  y •  Thus  =  -35-  x  "e  ^  210" ="5  '  *^^*  '^^  *^« 
If  is  multiplied  by  7  and  divided  by  6,  for  multiplying  the  denomina- 
tor divides  the  fraction  (lvii.  3) ;  but  a  fraction  may  be  multiplied  by 
7  by  dividing  its  denominator  (lvii.  3),  and  divided  by  4  by  dividing 

12  -^  6       2 
its  numerator  (lvi.  3) ;  thus:  -07-^^=—,  as  before.         A.  f. 

50.  Hence,  to  abbreviate  the  process  of  dividing  one  fraction  by 
another — Divide  the  numerator  of  the  dividend  hy  the  numerator  of 
the  divisor,  and  the  denominator  by  the  denominator,  if  it  can  he  done 
without  a  remainder  y  otherivise  the  quotient  will  he  a  complex  fraction. 

^      ,       120     ,      120       ^,  120    -f-  120  _  1 

51.  Divide -g3^  by  9^.     Thus:  -5307- -"90T  "T"    ^- ^^ 

52.  Divide  -ij  by  4;-f f  1  by  fj.  A.  f ;  H 

53.  At  -j^  of  a  dollar  a  yard,  how  many  yards  of  cloth  may  be 

15  ^   3        5 
bought  for  yf  of  a  dollar  1     -7^  ^  -7^  =  -j-=5.     Or  by  No.  35,  thus : 

15  ^  16        ^.  ,    ^  ,,.  ,  ,  15      ,      . 

Tfi"  X  T~'  "^"^^"  "y  cancelling  equal  terms  becomes  -«- ,  that  is, 

15^3=5,  as  before.  A.  5  yards. 

54.  Hence,  when  fractions  have  a  common  denominator,  the  pro- 
cess may  be  still  more  abbreviated,  thus : — Reject  the  common  de- 
nominator, then  divide  as  in  whole  numbers. 

bb.  How  many  times  greater  is  f  than  ^  ■?  A.  A  times. 

56.  How  many  times  is  -^  contained  in  \^\  A.  Q  times. 

57.  Divide  l||  by  3^^,  and  ^  by  i^.  A.  b;  3^^ 

58.  If  a  man  perform  3^^  of  his  journey  in  half  a  day,  how  many 
half  days  will  he  be  in  performing  | ,  of  the  journey  1  Jl.   12. 

59.  In  the  same  manner  we  might  divide  all  fractions,  if  we  reduce 

Q.  Divide  1|  by  4-  See  49.  How  and  when  may  the  process  be  still  more 
abbreviated?  54.  What  is  the  quotient  of  \^  divided  by  -A?  See  53.  Why 
not  use  the  denominators  at  all?  Can  all  fractions  be  divided  in  the  same 
manner?  59.    How  then  does  this  process  differ  from  the  general  rule  ?  60. 


142  ARITHMETIC. 

them  first  to  a  common  denominator;  for  example,  f  by  |.    Thus : 
f+|-fl^!f=l^=24^20-U.  A.   U- 

60.  This  process,  however,  involves  no  ncvi^  principle,  for  it  is  in 
reality  the  same  as  that  of  the  general  rule.  For,  by  inverting  the 
divisor,  the  same  terms  are  multiplied  together  in  both  cases. 

3  X  8  _  24 

61.  Thus,  taking  the  last  example,  -j-    ~K~~'n7{- 

62.  When  the  divisor  is  a  whole  number,  and  the  dividend  consists 
of  a  large  integer  and  a  fraction — Divide  each  separately,  and  if  the 
whole  number  leave  a  remainder,  multiply  it  by  the  denominator  of 
the  fraction,  and  add  the  product  to  tlie  numerator,  for  a  new  nume- 
rator, which  divide  as  before. 

63.  For  every  unit  of  the  whole  number  is  equal  to  as  many  parts 
of  the  fraction  as  are  indicated  by  the  denominator. 
3  )  4  6  8  2  4  8  64.  Divide  468,248  by  3,  4,  2,  5,  and  6. 
4)156082f           These  operations  involve  all  the  variations 

that  can  possibly  take  place  in  dividing  a 
fraction  by  an  integer. 

The  1st  rem.  is  f :  2nd,  2|=|^4=f : 
3d,  f-2=^:   4th,  K5=yV:   5th,  2yV= 

65.  Divide  47,325,737  by  6,  5,  4,  3,  and  2.  A.  65,730^1^. 

66.  Divide  76,543,210  by  8,  7,  5,  4,  and  3.  A.  22,780f||. 

67.  Divide  37,754,276  by  5,  4,  7,  6,  and  5.  A.  8,989yV^. 

68.  Divide  76,864,207  by  8,  7,  5,  5,  and  4.  A.  13,725^-^. 

69.  Divide  42,680,960  by  12,  2,  6,  2,  and  7.  A.  21,171^. 

70.  Divide  98,765,432  by  9,  8,  7,  6,  5, 4, 3,  and  2.  A.  ^12i^-^^. 

71.  Find  by  the  preceding  rules  what  number  multiplied  by  f  will 
make  15f.  A.  21. 

72.  What  part  of  108  is  y^  of  an  unit  ?  A.  ^^. 

73.  What  number  is  that  which,  if  multiplied  by  |  of  ^  of  15|, 
will  produce  only  |  of  an  unit  ?  A.  ^^. 


)  4  6  8  2  4  8 

)  1  5  6  0  8  2 

f 

2)39020 

f 

5)19510 

^ 

6)3902 

tV 

Jl.  6  5  0 

Ji 

MISCELLANEOUS    EXAMPLES. 

LXVII.  1.  A  gentleman  has  two  sons  ;  the  age  of  the  elder  added 
to  his,  make  126  years,  and  the  age  of  the  younger  son  is  equal  to  the 
difference  between  the  age  of  the  father  and  the  elder  son.  Now  if 
the  father  be  80  years  of  age,  how  old  is  each  of  his  sons? 

A.  34  years,  and  46  years. 

3.  What  number  is  that,  from  which,  if  a  twelfth  part  of  1,728  be 

Q.  How  may  a  mixed  number  be  divided  without  any  reduction,  by  a  whole 
number?  62.  Why  is  the  remainder,  if  there  be  any,  multiplied  by  the  denomi- 
nator of  the  fractional  part  ?  63. 


MISCELLANEOUS    EXAMPLES.  143 

deducted,  and  the  remainder  increased  by  the  ninety-fifth  part  of 
82175,  the  sum  will  be  1185 1  A.  464. 

3.  What  number  divided  by  1185  will  give  497  for  the  quotient, 
and  leave  just  a  fifth  part  of  the  divisor  remaining?     A.  589,182. 

4.  A  merchant,  having  bought  at  one  time  25bl.  27gal.  2qt.  Ipt. 
2gi.  of  molasses,  and  at  another  5  times  as  much,  sold  ^  of  the 
whole ;  how  much  has  he  on  hand,  unsold  1 

A.   116bl.  14gal.  Iqt.  3gi. 

5.  "At  the  Clinton  works  in  Scotland,  a  sheet  of  paper  has  been 
recently  [1839]  produced,  which,  though  but  50  inches  wide,  mea- 
sures a  mile  and  a  half  in  length."  How  many  square  feet  does  this 
mammoth  sheet  contain?  How  many  smaller  sheets  1^  feet  long 
and  1  foot  wide  will  it  make  1  How  many  quires  will  these  sheets 
make  1    What  will  be  their  value  at  84|  per  ream  ? 

A.  33,000  sq.  ft.;  26,400  sheets  ;   1,100  quires;  $247^. 

6.  How  many  acres  of  land  are  there  in  a  piece  80  rods  long  and 
36  rods  wide  1  A,  18A. — lu  a  piece  80|  rods  long  and  36^  rods 
wide?  A.   18A.  2R.  17||rd. 

7.  If  a  talent  of  silver  be  worth  jC357.  lis.  10|d.  sterling,  what  is 
the  value  of  a  shekel,  of  which  300  make  a  talent ;  and  what  is  the 
weight  of  a  talent,  a  shekel  weighing  9dwt.  3gr. 

A.  £i.  3s.  104^^d.;  1360Z.  17dwt.  12gr. 

8.  If  London  was  built  1108  years  before  Christ's  nativity,  how 
many  hours  is  it  since,  to  Christmas,  1835,  allowing  365}  days  to  the 
year?  ^  A.  25,798,338  hours. 

9.  A  father  gave  f  of  his  property  to  his  daughter,  4  to  his  son, 
and  the  rest,  being  $1,200,  to  his  nephew;  what  was  the  value  of 
the  father's  estate  ?  A.   $8,400. 

10.  "  The  tail  of  the  comet  of  1811  was  no  less  than  one  hundred 
and  thirty-two  millions  of  miles  in  length.  Now,  allowing  the  earth 
to  be  25,000  miles  in  circumference,  and  the  tail  of  the  comet  a  band- 
age, how  many  times  would  it  enwrap  the  earth  ?"  A.  5,280  times. 

11.  A  general  distributed  £307.  17s.  among  4  captains,  5  lieuten- 
ants, and  60  common  soldiers ;  to  every  lieutenant  he  gave  twice 
as  much  as  to  a  common  soldier,  and  to  every  captain  three  times  as 
much  as  to  a  lieutenant ;  what  did  each  receive? 

A.  Soldier  £3.  5|s.;  lieut.  £6.  lis.;  capt.  jClO.  13s. 

12.  What  number  is  that  from  which  if  you  subtract  xV  of  ^  of  a 
unit,  and  to  the  remainder  add  f  of  ^  of  a  unit,  the  sum  will  be  9  ? 

A       Q20S I 

13.  Suppose  a  quotient  to  be  3^  times  y\,  and  a  dividend  5|  times 
yj,  what  will  be  the  divisor?  A.  SyH. 

14.  A  merchant  invested  $8,300  in  a  certain  bank,  being  just  j-fy 
of  its  capital ;  what  was  the  capital  of  the  bank?       A.  $190,900. 

15.  A  merchant  gave  $1,956|  for  4  of  a  sloop,  and  f  of  the  value 
of  the  sloop  for  its  entire  cargo ;  what  was  the  estimated  value  of 
both  sloop  and  cargo  ?  A.  $4,794.03|. 


144  ARITHMETIC. 

16.  Suppose  a  man's  family  expenses  are  STSO/^  annually,  being 
only  I  of  his  profits  in  trade ;  how  much  then  can  he  save  every 
year?  ^  A.  $1,2172V 

17.  Suppose  a  carriage  wheel  to  be  16/^  inches  in  circumference, 
how  many  times  would  it  turn  round  in  going  367|  miles  1 

A.  1,530,534^1  times. 

18.  Divide  12  sq.  rd.  30  sq.  yd,  6  sq.  ft.  by  7,  and  multiply  1  sq. 
rd.  25f|  sq.  yd.  123?  sq.  in.  by  7. 

19.  Suppose  a  pile  of  wood  to  be  200|  feet  long,  6f  feet  high,  and 
4|  feet  wide ;  how  many  solid  feet  does  the  pile  contain,  and  how 
many  cords]  A.  6,194|  s.  ft.;  48C.  50f  s.  ft. 

20.  How  many  cubic  feet  in  a  box  9f  inches  long,  8}  feet  high, 
and  5  feet  widel  A.  411^  s.  ft. 

21.  Suppose  a  room  to  be  10  feet  between  floors  and  18y  feet 
square,  what  will  be  the  expense  of  plastering  it  at  8  cents  per  square 
yard?  There  are  4  sides  each  10  ft.  by  18|^,  and  the  surface  over 
head  is  18|  by  18^  A.  $9.44|1. 

22.  A  wealthy  merchant,  on  retiring  from  business,  invested  j\  of 
all  his  property  in  banks,  4  in  private  loans,  ^i  in  real  estate,  and  the 
rest,  which  was  83,000,  he  reserved  for  repairs  on  his  estate  ;  how 
much  must  he  have  accumulated?  A.  $216,000. 

23.  Suppose  a  man  buys  f  of  a  ship,  which  is  valued  at  $63,000, 
and  divides  it  equally  among  his  sons,  giving  to  each  -^f  of  his  part ; 
how  many  sons  has  he,  and  how  many  dollars  does  each  receive  ? 

A.  7  sons  ;  $6,750  each. 

24.  Divide  ff  of  a  roll  of  broadcloth,  which  contains  32  yards, 
into  equal  pieces,  each  to  contain  yj  of  the  whole  roll. 

A.  5 3^  pieces. 

25.  Suppose  two  boys,  having  bought  a  kite  together,  one  paying 
I  of  a  dollar  and  the  other  7  of  a  dollar,  sell  it  for  |  of  a  dollar  more 
than  they  paid  for  it ;  what  did  they  pay  for  the  kite  1         A.  $1|. 

■What  did  they  get  for  the  kite?  A.  $lf. 

What  is  each  one's  part  of  the  kite  ?  -4.  f ;  f . 

What  is  each  one's  share  of  the  profit?  A.  $^  ;  i^. 

What  is  each  one's  share  of  what  it  sold  for  ?      A.  $g4  ;  $1^- 

26.  If  ^  of  a  ship  valued  at  $20,000  be  worth  f  of  her  cargo,  what 
is  the  value  of  both  ship  and  cargo?  A.  $41,777^. 

27.  A  person  left  f  of  his  property  to  A,  fV  to  B,  I  to  C,  gV  to  D, 
^  to  E,  ^  to  F,  and  the  rest,  which  was  $800,  to  his  executor ; 
what  was  the  value  of  the  whole  property,  and  of  each  person's  share  1 

A.  A's  $4,000;  B's  $3,000;  C's  $1,250;  D's$500;  E'8$250j 
F's  $200. 


DECIMAL   FRACTIONS.  149 


DECIMAL    FRACTIONS. 

LXVIII.  1.  A  Decimal  FRACTioNMiffers  from  a  vulgar  fraction 
only  in  respect  to  its  denominator,  being  uniformly  either  10  or  100 
or  1000,  &c.,  and  therefore  it  need  not  be,  and  seldom  is,  expressed. 

2.  The  numerator  then  is  written  alone  with  a  point  before  it,  to 
distinguish  it  from  whole  numbers;  this  point  is  thence  called  a 
separatrix,  and  sometimes  the  decimal  point. 

3.  Thus  .3  is  1^  ;  .34  is  -^V  5  -345  is  -^%%',  .3450  is  tVotjV 

4.  Unity  then  in  decimals  is  first  divided  into  10  equal  parts, 
which  are  therefore  called  tenths. 

5.  The  TENTH  is  divided  into  10  other  equal  parts,  making  100 
equal  parts  of  unity,  which  are  thence  called  hundredths. 

C.  The  HUNDREDTH  is  divided  into  10  other  equal  parts,  making 
1000  equal  parts  of  unity,  which  are  thence  called  thousandths; 
and  so  on,  as  in  the  following 

TABLE    I. 

10-tenths make  1  unit. 

10-hundredths   -------    make  1  tenth. 

10-thousandths  -------    make  1  hundredth. 

10-ten-thousandths     -----    make  1  thousandth. 
10-hundred-thousandths  -    -    -    -    make  1  ten-thousandth. 
10-millionths      -------    make  1  hundred-thousandth. 

7.  Since  one  decimal  figure  has  for  its  denominator  1  with  one 
cipher,  as  .5=-^^\  two  decimal  figures,  1  with  two  ciphers,  as  .25= 
y^/o  ;  three  decimal  figures,  1  with  three  ciphers,  as  .125=y*^^^,  and 
so  on ;  therefore, 

8.  A  Decimal  Fraction  is  that  fraction  whose  denominator  is 
always  understood  to  be  a  unit,  or  1,  with  as  many  ciphers  annexed 
as  the  given  decimal  has  places  of  figures. 

9.  Thus,  .8  is  y\ ;  .08  is  y|o  ;  .35  is  yVo  ;  -0125  is  y^f^. 

10.  When  the  numerator  has  not  so  many  decimal  places  as  the 
denominator  has  ciphers,  we  must  prefix  ciphers  enough  to  the  nu- 
merator to  make  as  many. 

11.  Thus  yf  0  is  written  .05  ;  yo\T^=.0045 ;  yo4^-o-"-"-0006. 

12.  Since  .5=y^o>  •^5=y|o,  .005=y/oo5  then  .05  is  10  times  less 
in  value  than  .5,  and  .005  is  10  times  less  than  .05  : 

LXVIII.  Q.  How  does  a  decimal  fraction  differ  from  a  vulgar  one?  1. 
How  is  the  numerator  written  ?  2.  What  does  .3,  .34,  .345,  .3456,  with  the 
point  before  each  number,  mean?  3.  How  is  unity  divided  and  sub-divided  in 
decimals?  4,  5,  6.  Repeat  the  table  of  these  divisions.  What  is  the  de- 
nominator of  one,  two,  or  three  decimal  figures  ?  7.  What  then  is  a  Decimal 
Fraction  ?  8.  What  is  the  denominator  for  .8  ?  See  9.  What  is  the  denomi- 
nator for  .08  ?— for  .35?— for  .0125?  9.  When  are  ciphers  to  be  prefixed  to  the 
numerator?  10.     How  do  you  write  decimally  -S^,  or  y^VoTU  °^  ToT^cT  • 

1  A  Decimal  Fraction  is  so  called  from  the  Latin  word  decimus,  signifying  tenthi 
because  it  increases  and  decreasea  in  a  tenfold  proportion. 


146  ARITHMETIC. 

13.  For  10  times  yAs  is  TUii=Th,  and  10  times  jS^=:J>^%=^. 

14.  Hence,  a  cipher  placed  on  the  left  of  any  decimal  decreases  its 
value  in  a  tenfold  proportion,  by  removing  it  farther  from  the  separa- 
trix  or  decimal  point. 

15.  But  a  cipher  on  the  right  of  a  decimal  merely  changes  its  name 
without  altering  its  value. 

16.  For  .5  is  j%=\\  so  .50  is  ■^%=^,  and  .500  is  TFTnr=i  but 
the  first  is  read  5-tenths,  the  second,  50-hundredths,  the  third,  500- 
thousandths. 

17.  Decimals  then  increase  from  the  right  to  the  left  like  whole 
numbers,  and  of  course  their  decrease  from  right  to  left  is  in  the 
same  proportion. 

18.  Hence  every  removal  of  any  figure  one  place  further  towards 
the  right  decreases  its  value  in  a  tenfold  proportion. 

19.  Thus  555.555  is  really  500,  50,  5,  -^^,  ,fo,  two- 

20.  Our  system  of  notation,  then,  which  begins  in  whole  numbers, 
is  carried  out  by  means  of  decimals,  so  as  to  embrace  as  many  places 
below  units  as  above  or  beyond  them,  even  millions  and  miUionths, 
billions  and  billionths,  as  in  the  following 


g 


TABLE    II. 

CO 

§  g  I 

2  ^    M  H    2  2     2 

si  ^      ^      i       r^  S^ShSoSM 

i§iig||g    m^  ^  1 1  ^  ^  i  ^  ^ 

6555555555.55    5    555555 

Ascending.   "Cig  ^;^  Descending. 

21.  The  first  decimal  figure  is  5  tenths;  the  second  is  5  hun- 
dredths, or  the  first  two  55  hundredths ;  the  third  is  5  thousandths, 
or  the  first  three  555  thousandths ;  the  fourth  is  5  ten-thousandths, 
or  the  first  four  5555  ten  thousandths,  &c. 

RULE    FOR    NUMERATING    AND    READING    DECIMALS. 

22.  Begin  on  the  left  and  say,  tenths,  hundredths,  thousandths^ 
ten-thousandths,  dfc,  as  in  the  table. 

Q.  What  is  the  difference  in  value  between  ,5  and  .05?— .05  and  .005?  (See 
12.)  What  is  the  effect  of  the  cipher?  14,  What  is  the  use  of  a  cipher  on  the 
left  of  a  decimal?  15.  How  is  a  decimal  figure  affected  by  changing  its  place?  18. 
Why?  17.  What  is  the  value  of  each  figure  in  555.555?  19.  Describe  our  sys- 
tem of  notation.  20.  Repeat  the  decimal  part  of  the  table  beginning  with 
"tenths"  and  ending  with  "biUionths."  Suppose  each  decimal  place  to  be 
filled  with  the  figure  5,  what  would  be  the  value  of  the  first  5? — of  the  second?— 
third  ? — fourth  ?  &c.  21 .    What  is  the  rule  for  numerating  decimals  ?  22. 


DECIMAL    FRACTIONS.  147 

23.  Then  hegin  on  the  left  and  read^  giving  each  figure  the  value 
assigned  it  in  numerating. 

24.  Or  numerate  and  read  the  entire  decimal,  as  if  it  were  a  whole 
number,  giving  the  name  of  the  last  right  hand  place  to  the  whole. 

25.  Write  on  the  slate  the  decimal  figures  expressing  the  following 
numbers,  to  be  numerated  and  read  at  recitation. 

26.  Five  tenths. 

27.  Seventy-six  hundredths. 

28.  Nine  tenths  and  two  hundredths. 

29.  Three  hundred  and  twenty-one  thousandths. 

30.  Five  tenths,  two  hundredths,  and  six  thousandths. 

31.  Six  tenths,  two  hundredths,  three  thousandths,  and  one  ten 
thousandth. 

32.  Six  thousand  nine  hundred  and  fifteen  ten-thousandth. 

33.  Six  tenths,  1  ten  thousandth,  and  four  millionths. 

34.  Note. — Supply  all  vacant  places  with  ciphers. 

35.  Three  tenths,  five  thousandths,  and  two  millionths. 

36.  One  hundred  and  one  thousandths. 

37.  To  express  5  hundredths,  which  has  one  vacant  place,  viz. 
tenths,  we  prefix  1  cipher  [.05] ;  to  express  5  millionths,  which  has 
five  vacant  places,  we  prefix  5  ciphers  [.000005],  and  so  on  to  any 
extent. 

38.  Hence,  to  express  any  number  of  hundredths  or  thousandths, 
&c. — Prefix  as  many  ciphers  as  there  are  vacant  places  between  it 
and  the  separatrix. 

39.  Write  on  the  slate  and  recite  as  before  the  following  numbers. 

40.  Seven  hundredths. 

41.  Forty-five  ten-thousandths. 

42.  Six  hundred  thousandths  and  one  millionth. 

43.  Fifteen  hundred-thousandths  and  fifteen  billionths. 

44.  One  thousandth,  one  millionth,  and  one  billionth. 

45.  Nine  hundredths,  nine  thousandths,  and  nine  billionths. 

46.  Three  hundred  and  sixty-five  millionths. 

47.  One  hundred  and  twenty-five  trillionths. 

48.  When  a  whole  number  has  a  decimal  annexed,  they  form  a 
mixed  decimal  fraction,  and  may  be  read  like  decimals,  giving  the 
name  of  the  last  decimal  figure  to  both. 

49.  Thus  45.2  is  Ab^j,  or  V¥»  that  is,  452  tenths. 

50.  So  5.62  is  562  hundredths,  and  3.005  is  3005  thousandths. 

51.  In  the  examples  .5=y^  and  .25=y%*^,  if  we  annex  a  cipher  to 
.5  it  becomes  .50  =3^,  having  the  same  denominator  with  .25, 
therefore, — 

Q.  What  are  both  methods  of  reading  decimals  ?  23,  24.  How  are  6-tenths, 
1 -ten-thousandth  and  4-millionths  written  in  one  line?  34.  How  are  5-hun- 
dredths  or  5-millionths  written,  and  why?  37.  What  is  the  general  direction?  38. 
What  is  a  mixed  decimal?  48.  How  may  the  following  numbers  be  read,  viz. 
45.2,  5.62,  and  3.005?  [See  49,  50.] 


148  .    ARITHMETIC. 

52.  Whole  or  mixed  numbers*  and  pure  decimals  are  easily  reduced 
to  decimals  having  the  same  denominators  by  simply  annexing  ciphers. 

53.  Reduce  2.5,  8.1,  and  7.05,  each  to  hundredths. 

A.  2.50;  8.10;  7.05. 
64.  Reduce  Sj^jj,  17.8,  and  .212,  to  thousandths. 

A.  3.500;  17.800;  .212. 

55.  Reduce  the  following  numbers  to  decimals  having  the  same 
denominators.     8.5;  33-2^;  y-5_.  __?__.  75^3.  -^-L^;  981^^- 

56.  Answers.  8.50000  ;  3.21000  ;  .05000  ;  .00800  ;  756.30000 ; 
.00009;  981.10000. 

57.  In  decimals,  no  single  expression,  containing  any  number  of 
figures  whatever,  can  fully  equal  unity. 

58.  Thus,  .9  is  1-tenth  less  than  10-tenths,  which  make  one  unit ; 
so  .999999  is  .000001,  or  1  milhonth  less  than  1. 

59.  It  is  observable  also,  that  of  two  decimal  expressions,  the 
greater  one,  (no  matter  of  how  many  figures  either  consists)  has  the 
greater  number  of  tenths,  or  if  the  tenths  be  equal,  a  greater  number 
of  hundredths,  and  so  on. 

60.  Thus  .4  is  greater  than  .399999,  or  .3  with  any  number  of  9s 
that  can  possibly  be  annexed. 

61.  For  .4  is  (by  52)  =.4000000,  or  equal  to  .4  with  any  number 
of  ciphers  annexed ;  now,  .400000  is  obviously  greater  than  .399999. 


62.  Federal  Money,  by  assuming  the  dollar,  as  the  money  unit, 
is  perfectly  adapted  in  all  its  inferior  denominations,  to  the  decimal 
notation. 

63.  For,  as  10  dimes  make  one  dollar ;  10  cents  1  dime  ;  and  10 
mills  one  cent ;  dimes  are  lOths  of  dollars  ;  cents,  lOths  of  dimes  or 
lOOths  of  dollars,  and  mills  lOths  of  cents,  or  l,000ths  of  dollars. 

64.  Thus  $3,  2  dimes,  4  cents  and  5  mills  are  written  decimally 
$3,245,  that  is,  $3^"^%. 


REDUCTION   OF   DECIMALS. 

LXIX.  1 .  Reduction  of  Decimals  is  the  changing  of  their  forms, 
without  altering  their  value. 

CASE  1. 
To  reduce  a  decimal  fraction  to  a  vulgar  one. 

RULE. 
1.   Write  under  the  given  decimal  its  proper  denominator,  and  it 

Q.  How  may  whole  numbers,  or  decimals  of  different  denominators,  be  re- 
duced to  a  common  denominator?  52.  Reduce  2.5,  8,  and  7.05,  each  to  hun- 
dredths. 53.  Is  then  any  decimal  expression  fully  equal  to  unity  ?  What  is 
the  difference  in  value  between  unity  and  .9?  Unity  and  .999999?  Which  is 
the  greater  decimal,  .4  or  .399999?  60.  How  do  you  ascertain  it?  61.  What 
similarity  has  Federal  Money  to  decimals  ?  62. 


A. 

.  1 

A.  I 

;i 

A.h 

•^ 

A.^; 

A 

A         1     . 

1 

-*•    2F5-  5    ^BTToT- 

A'  -ffffo; 

61. 

^-  e^frr; 

5i. 

REDUCTION    OF    DECIMALS.  149 

becomes  a  vulgar  fraction,  ivhich  may  generally  be  reduced  to  lower 
terms. 

2.  Reduce  .5  to  a  vulgar  fraction.     -5  is  ^=|. 

3.  Reduce  .75  and  .125  to  vulgar  fractions. 

4.  Reduce  .875  and  .15  to  common  fractions. 

5.  Reduce  .05  and  .1875  to  common  fractions. 

6.  Reduce  .005  and  .0005  to  vulgar  fractions. 

7.  Reduce  .00125  and  6.25  to  vulgar  fractions. 

8.  Reduce  6.015  and  5.50  to  vulgar  fractions. 

CASE    II. 

To  reduce  a  vulgar  fraction  to  a  decimal. 

RULE. 

1.  Annex  a  cipher  to  the  numerator  and  divide  ly  the  denominator 
if  there  be  a  remainder,  annex  another  cipher  and  divide  as  before^ 
and  so  on  to  any  extent  required- 

2.  The  quotient  will  contain  as  many  decimal  places  as  there  are 
ciphers  annexed ;  but  if  there  he  not  as  many  places,  supply  the  defect 
by  prefixing  ciphers  to  the  quotient. 

3.  For,  annexing  one  cipher  to  the  numerator  multiplies  it  by  10, 
.  2  X  10=20  tenths.) 

which  brmgs  it  into  tenths,  (as  —  -r 

4.  Then  as  many  times  as  the  denominator  is  contained  in  the  nume- 

,  ^  ^  .     ,  .     ,     ^       .  20  tenths 

rator,  so  many  lOths  are  contained  m  the  fraction,  (as  -r-  =.4.) 

6.  Annexing  another  cipher  brings  the  numerator  into  hundredths, 
then  dividing  by  the  denominator  will  show  the  hundredths  contained 

in  the  fraction,  and  so  on,  (-}x  100=^^-  hundredths=.25.) 

6.  When  there  are  no  tenths,  hundredths,  &c.,  the  vacant  places 
in  the  quotient  must  be  filled  with  ciphers,  to  keep  the  significant 
figures  of  the  quotient  in  their  proper  places. 

7.  Reduce  ^  f ,  1,  oV'  '         ' 
2  )  1  .0      4  )3.00     8  )7.000     2  5  )  1  .00      2  0  0)1.000 

A.      .5     A.     .7  5      A.      .8  7  5        A.      .04  A.       .005 


8.  Reduce  gVo  3-^^  A\  *o  decimal  fractions.     A.  .255  ;  .0208. 

9.  Reduce  \  and  /j  to  decimal  fractions.  A.  .125 ;  .08. 

10.  Reduce  to  decimals  ^,  ^V?  i  ^A o)  2o4oo>  27o»  ^• 
Answers.    .5625;  .025;  .375;  .000625;  .00005;  .004;  .075. 

11.  Reduce  14|  to  a  decimal  fraction.     Reduce  the  fractional  part 
separately,  then  annex  it. A.  14.125. 

LXIX.     Q.  What  is  Reduction  of  Decimals  ?  1. 

Case  I.  Q.  What  vulgar  fraction  is  equal  to  .5  ?— to  .75  ?— to  .4  ?— to  .25  ? 
What  is  the  rule?  1. 

Case  II.  Q.  How  is  a  vulgar  fraction  reduced  to  a  decimal  one?  1.     How 
many  decimal  places  must  there  be  in  the  quotient?  2.  Why  is  the  cipher 
annexed  ?  3,  4.     Give  an  example.     Why  annex  two  or  more   ciphers  ?  5. 
Why  are  ciphers  in  some  instances  to  be  prefixed  to  the  quotient?  6. 
13* 


150  ARITHMETIC. 

12.  Reduce  to  decimals  9y,  5j,  6|,  5-^,  74520075-2^  and  ^. 

A.  9.2;  5.125;  6.875;  5.0625;  74520.00005;  .333333+. 
'  13.  Reduce  V  to  a  decimal.  V=5^==5.5,  or  11.0-^2  =  5.5,  re- 
collecting that  the  quotient  figure  or  figures,  before  ciphers  are  an- 
nexed, is,  of  course,  a  whole  number.  A.  5.5. 

14.  Reduce  to  decimals  Vg-V,  ^=/-S  Ifl,  »-V/-S  and  V- 

A.  5.555;  850.2;  30.25;  582.04;  3.33333+. 

15.  In  the  last  example,  the  decimal  will  repeat  33,  &c.,  for  ever, 
if  we  continue  the  operation. 

16.  Decimals  which  repeat  one  or  more  figures  are  called  Repeat- 
ing Decimals,  or  Repetends. 

17.  Repeating  Decimals  are  also  called  Infinite  Decimals; 
those  that  terminate,  or  come  to  an  end,  Finite  Decimals.* 

Q.  How  is  a  mixed  number  reduced  to  a  decimal?  11.  How  is  an  improper 
fraction  reduced  ?  13.  What  decimal  is  equal  to  51  ?— to  6^?— to  y  ?  What 
are  Repeating  Decimals?  16.     What  other  names  have  they,  and  when ?  17. 

*Repeatino  Decimals  are  also  called  Circulating  Decimals.  When  only  one 
figure  repeats,  it  is  called  a  single  repetend  ;  but  if  two  or  more  figures  repeat,  it  is  called 
a  compound  repetend :  thus,  .333,  &c.  is  a  single  repetend,  .010101,  &c.  a  compound 
repetend. 

When  other  decimals  come  before  circulating  decimals,  as  .8  in  .8333,  the  decimal  is 
called  a  mixed  repetend. 

It  is  the  common  practice,  instead  of  writing  the  repeating  figures  several  times,  to 
place  a  dot  over  the  repeating  figure  in  a  single  repetend;  thus,  .111,  &c.  is  written  1; 
also  over  the  first  and  last  repeating  figure  of  a  compound  repetend ;  thus,  for  .030303, 
&c.  we  write  .03. 

The  value  of  any  repetend,  notwithstanding  it  repeats  one  figure  or  more  an  infinite 
number  of  times,  coming  nearer  and  nearer  to  a  unit  each  time,  though  never  reaching  it, 
may  be  easily  determined  by  common  fractions  ;  as  will  appear  from  what  follows. 

By  reducing  xr  to  a  decimal,  we  have  a  quotient  consisting  of  .1111,  &c.,  that  is,  the 
repetend  1 ;  since  ^  is  the  value  of  the  repetend  1,  the  value  of  .333,  &c.,  that  is,  the 
repetend  3,  must  be  three  times  as  much ,  that  is,  -gr  and  4=?y;  5  =^;  and  9  =|-=1 
or  the  whole. 

Hence  we  have  the  following  Rule  for  changing  a  single  repretend  to  its  equal  com- 
mon fraction;— Make  the  given  repetend  a  numerator,  writing  9  underneath  for  a  de- 
nominator, and  it  is  done. 
What  is  the  value  of  .i?  Of  .2?  Of  .4?  Of  .7?  Of  .8?  Of  .6?     A.  ^,  f,  f,  ^,  f,  f. 
By  changing  g-g  to  a  decimal,  we  shall  have  .010101,  that  is,  the  repetend  ,'o'l.    Then, 
the  repetend  .04,  being  4  times  as  much,  must  be  ^,  and  . 36  must  be  f  f ,  also,  .45 = ^g- 
If  ir§^  be  reduced  to  a  decimal,  it  produces  .00 i.    Then  the  decimal  .004,  being  4 
times  as  much,  is  g^,  and  .036=^^^.    This  principle  will  be  true  for  any  number  of 
places. 

Hence  we  derive  the  following  Rule  for  reducing  a  circulating  decimal  to  a  common 
Auction  : — Make  the  given  repetend  a  numerator ;  and  the  denominator  will  be  as  many 
9s  as  there  are  figures  in  the  repetend. 
1. 

7  2 

Change  .72  to  a  common  fraction.    A.  g^ — yy 
Change  .003  to  a  conmion  fraction.    A.  T51i~7TS- 


REDUCTION    OF    DECIMALS.  151 

18.  In  general,  whether  the  decimal  be  finite  or  infinite,  three  or 
four  places  are  sufiicient  for  most  practical  purposes. 

19.  Change  j^  to  a  decimal  fraction.  A.  .1111-f-or.llll^. 

20.  Change  f  to  a  decimal  fraction.  A.  .6666+  or  .666f. 

21.  Change  3^2  to  a  decimal  fraction.  A.  .0937.+ 

22.  Change  SyV  to  a  decimal  fraction.  A.  8.062y5g. 

23.  Change  $|  to  cents  and  mills.  A.  $.125  =  .12/o==12|. 

24.  Change  $8yV  to  dollars  and  cents.        A.  $8.3125=S8.3l|. 

25.  Change  £50^  to  a  decimal  form.  A.  £50.625. 

26.  Change  3^j  miles  to  a  decimal  form.  A.  3.04m. 

27.  Reduce  to  single  fractions  first  and  then  to  decimals  •§  of  f ; 

|of|;-|and^\.  A.  .3333^;  .46875;  4.5;  .153846yV 

3  ^4 

CASE    III. 

To  reduce  a  simple  number  of  a  given  denomination  to  a  decimal 
of  a  higher  denomination. 

RULE. 
1.  Divide  as  in  Reduction  of  whole  numbers,  annexing  ciphers  and 
pointing  off  the  places  for  decimals  in  each  quotient,  as  in  the  last 
Case,  and  for  the  same  reasons. 

4  )  3  .  0  0  qr.  2.  Reduce  3  farthings  to  the  decimal 

of  a  pound. 

3.  Here  3qr.-^4qr.=.75of  apenny-r- 
12d.=.0625  of  ashil.-^20s.  =  .003125 
of  a  pound.  A.  £.003125. 

Q.  How  many  places  of  decimals  are  generally  sufficient  ?  18.  What  are 
Circulating  Decimals  ?  [See  reference  from  17.]  What  are  single,  compound, 
and  mixed  repetends?  Of  the  decimals  .333,  &c.,  .0101,  &c.,  and  .8333,  &c., 
which  are  the  repeating  figures  ?  What  is  the  proper  name  for  each  of  these 
decimals?  How  are  repetends  distinguished  from  other  decimals?  How 
is  the  value  of  the  repetend  .3  expressed,  and  why?  What  is  the  rule  for  it? 
What  is  the  value  of  .2  .7  and.  .01  ?  What  is  the  rule  for  reducing  a  circulating 
decimal  to  a  vulgar  fraction  ?     Reduce  to  a  vulgar  fraction  .18,  .72  and  .003. 

Describe  the  process  of  finding  the  value  of  the  mixed  repetend  .83.     What  is 
the  rule  ? 

In  the  following  example,  viz.  Change  .83  to  a  common  fraction,  the  repeating  figure  is 
3,  that  is,  f,  and  .8  is  yq-  ;  then  -g,  instead  of  being  f  of  a  unit,  is,  by  being  in  the  second 
place,  ^  of  i-o-=^o^;  then  jo^  andg-V  added  together,  thus,  tV"^¥g  "i^'^f'o.  -Ans. 

Hence,  to  find  the  value  of  a  mixed  repetend— First  find  the  value  of  the  repeating 
decimals,  then  of  the  other  decimals,  and  add  these  results  together. 

Change  .916  to  a  common  fraction.    A.  Too^"^! 
=  .916. 

Change  .203  to  a  common  fraction.    A.  _M_. 

To  know  if  the  result  be  right,  change  the  common  fraction  to  a  decimal  again.  If  it 
produces  the  same,  the  work  is  right. 

Repeating  decimals  may  be  easily  multiplied,  subtracted,  &c.  by  first  reducing  them 
to  their  ec^ual  common  fi-actions. 


12). 

,  7  5  0  0  d. 

2  0), 

,062500s. 

£ 

.003125 

'1^  ARITHMETIC. 

4.  Reduce  35  rods  to  the  decimal  of  a  mile.         A.  .109375m. 

5.  Reduce  9  pence  to  the  decimal  of  a  pound.  A.  £.0375. 

6.  Reduce  2  quarters  to  the  decimal  of  a  ton.  A.   .025T. 

7.  Reduce  8  drams  to  the  decimal  of  a  ton.      A.  .000015625T. 

8.  Reduce  3  gills  to  the  decimal  of  a  hogshead. 

A.  .001488^hhd. 

9.  Reduce  2%  nails  to  the  decimal  of  a  yard.  The  2ina.  ==2.375 
na.-J-4na.=.59375qr.  &c.  A.  .1484375yd. 

10.  Reduce  4|  pence  to  the  decimal  of  a  shilling.      A.   .40625. 

11.  Reduce  9}  shillings  to  the  decimal  of  a  dollar,  (6s.)  that  is,  to 
dollars  and  cents.  A.  $1.541f. 

12.  Reduce  15f  shillings  to  dollars  and  cents.  A.  $2,625. 

13.  Reduce  26f  hours  to  the  decimal  of  a  day.  A.  1.1  day. 

CASE    IV. 

To  reduce  compound  numbers  to  decimals  of  higher  denominations. 

RULE. 

I.  Divide  as  in  the  last  Case.,  annexing  each  decimal  quotient  to 
the  integer  of  that  denomination. 

2.  Reduce  £9.  10s.  6d.  3qr.  to  the 
decimal  of  a  pound. 

Recollect  to  make  the  decimal  pla- 
ces in  the  quotient  equal  to  those  in 
the  dividend,  according  to  Case  III. 

3.  Reduce  £5.  lis.  4*  pence  to  the  decimal  of  a  £. 

A.  £5.56875. 

4.  Reduce  7s.  6d.  3qr.  to  the  decimal  of  a  £.        A,  jC.378125. 

5.  Reduce  8oz.  17dwt.  to  the  decimal  of  alb.  A.  .73751b. 

6.  Reduce  3qr.  3  na.  to  the  decimal  of  a  yd.  A.  .9375yd. 

7.  Reduce  2qr.  2na.  to  the  decimal  of  an  E.  e.  A.  5.  E.  e. 

8.  Reduce  £8.  17s.  6id.  to  the  decimal  of  a  £.     A.  £8.878125. 

9.  Reduce  5T.  2cwt.  3qr.  71b.  to  the  decimal  of  a  ton. 

A.  5.141T. 

10.  Reduce  3.5  shillings  to  the  decimal  of  a  £.  A.  £.115. 

II.  Reduce  12h.  15.3m.  to  the  decimal  of  a  day.    A.  .510625d. 
12.  Reduce  2m.  4fur.  20rd.  4yd.  1ft.  2  in.  l^b.c.  to  the  decimal 

of  a  league.  A.  .8551. 

CASE   V. 

To  reduce  a  decimal  of  a  higher  denomination  to  integers  of  lower 
denominations. 

Case  III.  Q.  How  are  3  farthings  reduced  to  the  decimal  of  a  pound  ?  3. 
What  is  this  case?  [See  Case  iii.]  What  is  the  rule?  1.  What  decimal  of  a 
pound  is  15  shillings  ? — is  7  shillings  ? — is  3  shillings  ?  What  decimal  of  a  mile 
is  32  rods  ? — is  16  rods  ? 

Case  IV.  Q.  How  do  you  divide  farthings,  pence,  &c.  to  reduce  them  to 
the  decimal  of  a  pound?  2.  What  is  this  case  ?  [See  the  Case.]  What  is  the 
rule?  I 


4  )  3 

12)6. 

.   0  0  qr. 
7  5  0  0  d. 

2  0)10, 

.56250s. 

A.  £  9 

.528125. 

ADDITION    OF    DECIMALS.  153 

1.  This  Case  is  the  reverse  of  the  last  and  proves  it. 

RULE. 

2.  Multiply  as  in  Reduction  of  ivhole  numbers,  and  make  the  deci- 
mal places  in  each  product  equal  to  those  of  the  multiplicand ;  then 
the  several  excesses  on  the  left  will  constitute  the  compound  number 
required. 

3.  What  is  the  value  of  .26  of  a  shilling  ? 

.26s.  For,  .26s.  =  ^^^  x  12d.  =  ^Ifd.  =  SyVo  =  3.12, 

1  2  d  .       that  is,  as  .26  has  two  decimal  places,  so  must 
tbe  product  have  two. 

4.  What  is  the  value  of  .845  of  a  £  1 
Here,  £.845  X  20s.  =  16. 900s.,  making  three 
decimal  places,  because  £.845  has  three ; — • 
next  multiply  only  the  .900s.  ;  because  the 
16s.  is  an  integer,  and  therefore  a  part  of  the 
answer.     Do  the  same  with  the  .800d. 

5.  What  is  the  value  of  .645  of  a  £  1  A.  12s.  lOd.  3]qr. 

6.  What  is  the  value  of  .375  of  a  £  1  A,  7s.  6d. 

7.  What  is  the  value  of  .7375  of  a  lb.  ?  A.  8oz.  17dwt. 

8.  What  is  the  value  of  .9375  of  a  yd.  1  A.  3qr.  3na. 

9.  What  is  the  value  of  .025  of  a  mile  1  A.  8rd. 

10.  What  is  the  value  of  .0125  of  a  mile  ]  A.  4rd. 

11.  What  is  the  value  of  .025  of  a  ton?  A.  2qr. 

12.  What  is  the  value  of  .0025  of  a  £  ]  A.  ^.j^qr.  =2fqr. 

13.  What  is  the  value  of  .005  of  an  hour  ?  A.   18sec. 

14.  What  is  the  value  of  005  of  a  year  ?  A.  Id.  19h.  48m. 

15.  Change  .00025T.  of  round  timber  to  inches.        J..  21.6  in.     • 

16.  How  many  quarters  in  .534575yd.?  A.  2  qr.  yg^^\na,. 

17.  What  compound  number  will  express  .042675  of  a  year  1 

A.  15d.  13h.  49m.  58y^Tysec. 


Ans.3 

.  1  2  d  . 

£ 

.8  4  5 
2  0  s. 

1  6 

.900s. 
1  2d. 

1  0 

.  8  0  0  d. 
4  qr. 

3  . 

2  0  0  qr. 

ADDITION   OF   DECIMALS. 

LXX.  1.  Since  decimals  increase  towards  the  left  like  whole 
numbers,  like  them,  therefore,  they  may  be  added  and  subtracted, 
multiplied  and  divided. 

Case  V.  Q.  What  is  the  value  of  .26  of  a  shilling?  3.  Why  point  off  two 
decimal  figures  in  the  quotient?  3.  What  is  the  rule?  2.  What  is  the  value 
of  .5  of  a  pound? — .75  of  a  ewt.? — .75  of  a  shilling? — .15  of  a  furlong? 

LXX.  Q.  How  are  decimal  operations  performed,  and  why?  1.  What  is 
the  rule  for  Addition  of  Decimals  ?  3.  What  is  the  sum  of  .2  .5  and  .25  ?  Add 
together  4.5  yards  and  5.5  yards. 


154  ARITHMETIC. 

2.  The  only  difficulty  attending  these  operations  consists  in  ascer- 
taining where  the  decimal  point  should  stand.  This  will  be  noticed 
in  its  proper  place. 

RULE. 

3.  Write  tenths  under  tenths,  hundredths  under  hundredths,  djfC.j 
and  add  as  in  whole  numbers.  Then  make  the  places  for  decimals  in 
the  answer  equal  to  the  greatest  number  of  decimal  places  in  any  of 
the  given  numbers. 

4.  Add  into  one  sum  .5;  .875;  and  .25:  and  into  another  sum 
62.25,  350.009  and  .0036. 

•  ^  „  Add  the  numbers  as  they  ^  ^  n     0  0  Q 

•  2  -         stand ;  thus  5  is  5 ;  5  and  7  0  0  3  6 

A.  fT6T"5    "-'^  '^'  "='"'''"«  ^'  *"■  A.  4  12'.26a6 

5.  If  in  adding  .875,  .25  and  .5  we  reduce  them  all  to  lOOOths,  the 
proper  denominator  for  .875,  we  have  -^^o^  tofo-»  ^"^^  tfof?  whose 
sum  is  tII4=1iWo^— 1-625,  hence  the  reason  for  pointing  off  as  the 
rule  directs. 

6.  Add  together  2.35C. ;  450.009C. ;  .0839C. 

A.  452.4429  cords. 

7.  Add  together  3. 5689T.;  245.003T. ;  6.8T. 

A.  255.3719  tons. 

8.  Add  together  .0632 ;  .08;  .456;  .81  and  15.      A.  16.4092. 

9.  Add  together  $16,375;  $81,065  and  25  cents.       A.  $97.69. 

10.  Add  together  8.5  ;  834.6758  ;  8.35;  4236.2.  A.  5087.7258. 

11.  Write  decimally  and  add  2  dollars  and  30  cents;  4  dollars,  9 
dimes,  and  8  cents ;  7  dollars  and  8  mills ;  3  dimes,  7  cents ;  9  dimes 
and  2  mills.  A.  15.56. 

12.  Write  decimally  and  add  45y\hhd.,  68i4ohhd.,  96y||ohhd., 
and8y|ohhd.  A.  217.395hhd. 

13.  Find  the  sum  of  6  tenths,  35  hundredths,  6  thousandths,  and 
35  hundred-thousandths.  A.  .95635. 

14.  What  is  the  amount  of  555  and  5  thousandths ;  5  and  505 
millionths;  620  ten-millionths  ;  36  billionths  ;  428  and  15  ten-thou- 
sandths ;  5  million  and  5  millionths  T         A.  5000988.007072036. 

15.  Reduce  first  to  a  decimal  fraction  by  Case  ii.,  then  add  the 
following  numbers,  viz. — 45|  yards;  10y\  yards;  67|  yards;  and 
92V  yards.  A.  132.4775. 

'16.  Add  into  one  sum  |,  y^^,  f,  and  |.  A.   1.8375. 

17.  What  is  the  sum  of  ^^0,  y^o)  ilh,  tHwj  to5  rmlo,  and 
.000005^  A.   .692065. 

18.  Reduce  the  following  numbers  to  a  decimal  of  the  highest  de- 

Q.  What  is  the  sum  of  .3,  .25,  .2,  and  .25?  How  are  vulgar  fractions,  when 
connected  with  whole  numbers,  added?  15.  What  mixed  decimal  will  express 
the  sum  of  2i  and  2|  ?    How  are  compound  numbers  added  decimally  ?  18 


SUBTRACTION    OP    DECIMALS.  15^ 

nomination  mentioned  in  either,  then  add  them  together,  viz. — 10s. 
3d.,  £3.  15s.,  £9.  4s.  6d.  A.  jei3.4875. 

19.  Add  together  6yd.   Iqr,,  5qr.  3  na.,  17|yd.,  and  256yd.  2|qr. 

A.  281.8625  yards. 

20.  Find  the  sum  of  60|  years,  196f  years,  675f  years,  45y'V  years, 
and  20|  years.  A.  998.3375  years. 

21.  Add  decimally  19  miles,  3  furlongs,  20  rods ;  31  miles,  5  fur- 
longs; and  98  miles,  2  furlongs,  15  rods.       A.  149.359375  miles. 


SUBTRACTION   OF  DECIMALS. 

RULE. 

LXXI.  1 .  Write  tenths  under  tenths,  hundredths  under  hundredths^ 
^c.  ;  then  subtract  as  in  whole  numbers  and  point  off  as  in  Addition. 
2.  Suppose  a  merchant  bought  14.25  barrels  of  flour,  and  sold 
8,375  barrels ;  how  many  barrels  has  he  on  hand  1 

14.25  3.  Say,  5  from  10  leaves  5,  1  to  carry  to  7. 

^  >  3  7  5        For   U.25  =  Ut^%%-8^'^  (by  lxiv.  27.)= 
5,875        5,875  barrels,  Ans. 

4.  From  $90,025  take  S8.6285.  A.  S81.39G5. 

5.  From  $38,036  take  S4.0375.  A.  $33.9985. 

6.  Bought  513.025  barrels  of  flour,  and  sold  95.0375  barrels;  how 
much  had  I  left  ?  A.  417.9875  barrels. 

7.  From  891.037  take  89.0738.  A.  801.9032. 

8.  From  376,683  take  47.0931.  A.  329.5899. 

9.  From  83.12  take  5.3758.  A.  77.7442. 

10.  From  835.2  take  .1234567.  A.  835-0705433. 

11.  Subtract  $53,008  from  $100.  A.  $46,992. 

12.  From  $5  take  5  dimes  and  5  mills.  A.  $4,495. 

13.  From  8  take  9  thousandths.  A.  7.991. 

14.  From  1  take  1  hundredth.  A,  .99 

15.  From  1  take  1  millionth.  A.  .999999. 

16.  How  much  greater  is  4  than  3.99999999?    A.  .00000001. 

17.  How  much  smaller  is  .999999  than  U  A.   1  millionth. 

18.  How  much  less  than  1  is  1  trillionth? 

A.  .999999999999 

19.  From  191451  take  7918|.  A.  11227.375. 

20.  From  $89f  take  $37,125.  A.  52.275. 

21.  From  17|  barrels  take  92^  barrels.  A.  8.475bl. 

22.  From  £95^  take  18|s.  [See  Case  iv.]  A.  £94.185. 

23.  From  408^^  acres  take  329|  rods.  A.  406.015A. 

LXXI.  Q.  What  is  the  rule  for  subtracting  decimals?  1.  What  is  the  dif- 
ference between  .3  and  .25? — between  1  and  .1? — between  .5  and  .18? — be- 
tween 2.5  and  3 ? — between  .9  and  unity? 


156  ARITHMETIC. 

24.  From  10  eagles  and  ^^  of  a  mill  take  $99.9998.      A.  -j^m. 

25.  From  an  unit  subtract  5  millionths.  A.  .999995. 

26.  From  357 g^lb.  take  22, 4:flb.  A.  335.132251b. 

27.  From  £8,  173.  efd.  take  £3.  19s.  6^%d.  A.  £4.901. 


MULTIPLICATION  OF  DECIMALS. 

LXXII.  1.  Every  decimal  has  as  many  places  as  its  denominator 
has  ciphers,  as  y\  is  .5 ;  j-f^  is  .05,  &c. 

2.  The  product  of  any  two  decimals,  then,  must  have  as  many  de^ 
cimal  places  as  the  product  of  their  denominators  has  ciphers. 

3.  Thus,  .5x.7=.35,  for  AXto=t¥t=.35. 

4.  The  product  of  the  denominators  of  any  two  decimals  has  as 
many  ciphers  as  both  its  factors,  (xviii.) 

O.    inus,  100 '^To"~Tooir5  ^•"^  ToolTo '^To~~Tooooo' 

6.  Hence  the  product  of  any  two  decimal  expressions  must  have  as 
many  decimal  places  as  both  its  factors. 

7.  Thus.5x.87=.435;  for  y^y  ^tVo=T()¥o=-435. 

8.  When  the  product  has  not  as  many  places  of  figures  as  its  fac- 
tors have  places  of  decimals,  we  must  supply  the  deficiency  by  prefix- 
ing ciphers. 

9.  Thus,  .2x.4=.08  because  t^oXA=tfo  = -08. 

GENERAL    RULE. 

10.  Multiply  as  in  ivhole  numbers,  and  point  off  so  many  places  for 
decimals  in  the  product  as  are  equal  to  the  decimal  places  in  both  the 
factors ;  but  if  the  product  has  not  so  many  places,  prefix  ciphers  to 
make  up  the  number. 

1.  Multiply  4  5.625  Here  only  one  factor  is  a  decimal, 

by 5        and  it  has  3  places ;  therefore  make 

A.  228.125        3  decimal  places  in  the  product. 

12.  Multiply  81.235  wine  gallons  by  35.  A.  2843.225. 

13.  Multiply  90,325  puncheons  by  .45.  A.  40,646.25. 

14.  Multiply  3.251  Here  are  4  decimal  places  in  both 

by ^        the  factors,  therefore  make  4  decimal 

-^*  2.2757        places  in  the  product. 

15.  Multiply  2.345  ale  gallons  by.l5.  ^.  .35175  of  an  ale  gallon. 

LXXII.  Q.  How  is  yf^  written  decimally?  What  is  the  rule  for  it ?  1. 
"VMiy  does  .7  multiplied  by  .5  make  .35?  4.  What  is  the  rule  for  ascertaining 
the  decimal  places?  2.  How  many  ciphers  have  all  such  products?  4.  What 
is  the  inference  in  respect  to  pointing  off  the  product  in  decimals  ?  6.  When 
are  ciphers  to  be  prefixed  ?  8.  Why  then  is  .08  the  product  of  .2  by  .4  ?  9. 
General  Rule?  10.  Multiply  .8  by  .6,  by  .06,  by  .006,— .8  by  .5,  by  .05,  by 
0005 ;— .08  by  .08  by  .008. 


MULTIPLICATION    OF    DECIMALS.  157 

16.  Multiply  75.06  beer  gallons  by  .19.        A.   14.2614  gallons. 

17.  Multiply  .113  5  „    ^     .     •  u        •        .u    .       ^ 

jjy  3  Prefix  1  cipher,  since  the  two  fac- 


.A.  .  0  3  4  0  5 


tors  have  5  decimal  places.  See  8. 


18.  Multiply  .085  of  a  dollar  by  .39.  A.  $.03315. 

19.  Multiply  .009  of  a  gallon  by  .05.         A.  .00045  of  a  gallon. 

20.  What  will  3719.25  needles  cost  at  $.005  a-piece  1 

A.  $18.59ct.  e^^Vm- 

21.  Multiply  .618  of  a  hogshead  by  .312.  A.  .192816hhd. 

22.  Multiply  .521  of  a  bushel  by  .48.  A.   .25008bu, 

23.  Multiply  .235  of  a  century  by  .45.  A.   .10575C. 

24.  Multiply  .375  of  a  square  inch  by  .00027.  A,  .00010125sq.in. 

25.  Multiply  8.165  of  a  minute  by  .00089.      A.  .00726685min. 

26.  What  will  800  trees  cost  at  $.375  a-piece?  A.  $300. 

27.  Multiply  800  and  .008  together.  A.  6.4. 

28.  Multiply  5  and  .0005  together.  A.  .0025. 

29.  Multiply  .16  and  500  together.  A.  80. 

30.  Multiply  .003  by  8500000.  A.  25500. 

31.  Since  decimals  increase  from  the  right^c^the  left  in  a  tenfold 
proportion,  therefore, — 

32.  To  multiply  by  10,  or  100,  or  1,000,  &c. — Merely  remove  the 
separatrix  one  place  farther  towards  the  right  for  every  cipher,  and 
it  is  done. 

33.  Multiply  .3621  by  10;— by  100;— by  1,000;— by  10,000 ;— by 
100,000.  A.  3.621;  36-21;  362.1;  3621;  36210. 

34.  What  would  one  million  of  flax  seeds  cost  at  $.000001  for 
each  seed?  A.  $1. 

35.  Multiply  25  millionths  by  18  thousandths.     A.  .000000045. 

36.  Multiply  .02562  into  12J-.  ^1.  .31598. 

37.  What  will  be  the  cost  of  333^  Rohan  potatoes  at  $-0645 
a-piece  1  A,  $21^. 

38.  What  will  415]^  barrels  of  sugar  weigh,  the  average  weight  of 
each  being  495.00025  pounds'?  A.  205,524.1038. 

39.  Reduce  the  fractional  parts  of  the  following  numbers  to  deci- 
mals, then  multiply  them  together,  viz.  30|  by  5|.        A.  166.375. 

40.  Multiply  decimally  ^0\  sq.  yd.  by  9  sq.  ft.  A.  272.25  sq.  ft. 

41.  Multiply  5.5  by  b\  as  a  common  fraction.      A^  30.25 =30^. 

42.  Multiply  5|  by  5f  decimally.  A.  30.25. 

43.  What  will  18^  hogsheads  of  molasses  cost  at  the  rate  of  ISgg- 
dollars  per  hogshead  ]  A.  $284.0625." 

44.  What  will  7i  loads  of  hay  cost  at  £Z.  10s.  Od.  per  load? 
Reduce  both  to  decimals  by  lxix.  Case  iv.  A.  £26AZ75. 

Q.  How  are  decimals  multiplied  by  10  or  100,  &c.  easily?  32.  Why  has  the 
process  this  effect?  31.  What  is  the  product  of  .1234  multiplied  by  10? — by 
100?— by  1,000? 

14 


158  ARITHMETIC. 

45.  Multiply  £2.  3s.  8|d.  by  5  decimally.  A.  jClO.921875. 

46.  Multiply  2m.  Gfur.  30]rd.  by  5..  A.  14.221875  miles. 

47.  What  will  10  tons  15cwt.  of  Russia  iron  cost  at  £2.  3s.  6d. 
per  toni     The  product  is  JC23.38125.  (lxix.  Case  v.) 

A.  £23.  7s.  7ld. 

48.  What  will  14  hogsheads  18.9  gallons  of  molasses  cost  at  £3. 
15s.  9d.  per  hogshead  1  A.  £54.  3s.  2id.+ 

49.  Suppose  a  certain  farm  to  consist  of  200A.  3R.  25rd.;  what 
will  be  its  value  at  $25,375  per  acre.  A.  S5097.996.+ 

50.  If  a  man  travels  30m.  3fur.  15|rd.  a  day  for  26^  days,  how  far 
will  he  have  traveled  during  that  time  1        A.  806m.  1  fur.  30^rd. 

51.  WTiat  will  lOf  bales  of  cotton  cost,  each  bale  weighing  3c'wi;. 
2qr.,  at  810.62^  per  cwt.?  A.  $399.76|+, 

52.  How  many  solid  feet  in  a  stick  of  timber  40ft.  9in.  long,  1ft. 
3in.  wide,  and  1ft.  9in.  deep?     What  will  it  cost  at  25cts.  a  foot? 

A.  89ft.  243in.  nearly;  $22,285+. 


DIVISLON    OF   DECIMALS. 

LXXIII.  1.  We  have  seen  that  the  factors  in  multiplication  be- 
come the  divisor  and  quotient  in  division,  xxi.  53. 

2.  In  multiplication  of  decimals,  the  product  has  as  many  decimal 
places  as  both  its  factors. 

3.  In  division  of  decimals,  therefore,  the  divisor  and  quotient  must 
have  as  many  decimal  places  as  the  dividend. 

4.  Thus  .7x.5=:.35,  product:  then,  .35-^.5=. 7,  quotient. 

5.  The  same  result  will  follow  from  considering  the  decimals  as 
vulgar  fractions. 

6.  Thus  .5  is  ^  or  j%\;  then  -f3^--^5_o_(by  lxvi.  54)-?4=.7, 
quotient. 

7.  From  the  above  it  follows,  that  when  the  divisor  is  a  whole 
number,  the  quotient  alone  has  as  many  decimal  places  as  the  divi- 
dend. 

8.  And  when  the  divisor  alone  has  as  many  decimal  places  as  the 
dividend,  the  quotient  is  a  whole  number. 

GENERAL    RULE. 

9.  Divide  as  in  whole  numbers,  and  point  off  figures  enough  t9 
make  the  decimal  places  in  the  divisor  and  quotient  just  equal  to  those 
in  the  dividend,  prefixing  ciphers  to  the  quotient  when  necessary  to 
make  out  the  number. 

LXXIII.  Q.  How  many  decimal  places  has  any  product?  2.  How  many 
have  the  divisor  and  quotient?  3.  Why?  1.  Why  is  .7  the  quotient  of  .35 
divided  by.  5  ?  6.  When  does  the  quotient  have  as  many  decimals  as  the  divi- 
dend? 7.  When  is  the  quotient  a  whole  number?  8.  What  is  the  general 
rule?9,  10.  11. 


DIVISION    OF    DECIMALS.  159 

10.  When  the  dividend  has  not  so  many  decimals  as  the  divisor, 
make  them  first  equal  by  annexing  ciphers,  in  which  case  the  quotient 
will  be  a  whole  number. 

1 1 .  When  any  dividend,  either  with  or  without  ciphers  annexed, 
as  above,  does  not  contain  the  divisor,  or  when  there  is  a  remainder, 
annex  ciphers,  and  for  every  such  cipher  reckon  another  decimal  place 
in  the  quotient. 

12.  What  is  the  quotient  of  .75  divided  by  5  ? 

5  )  .  7  5  When  the  divisor  is  an  integer,  the  quotient  has 

A.         .   1  5        as  many  decimals  as  the  dividend.  (See  above,  7.) 

13.  Divide  87.745  barrels  among  7  persons.  A.  12. 535. 

14.  Divide  9.031455  into  9  equal  parts.  A.  1.003495. 

15.  What  is  the  quotient  of  .75  divided  by  .25  ? 

.  3  5  )  .  7  5         When  the  divisor  has  as  many  decimals  as 

A 5     the  dividend,  the  quotient  is  a  whole  number. 

^- 1     (See  8.) 

15.  How  many  times  .108  in  .972?  A.  9  times. 

16.  How  many  times  .00009  in  .00045 1  A.  5  times. 

17.  When  8  barrels  of  flour  cost  $80.75,  what  does  one  barrel  cost  ? 
8  )80.  75000     See  the  rule  (11)  for  annexing  ciphers 

$10.0937  5  =  $10.  9c.  3yVoni.  Answer. 

18.  Divide  $50.5  into  16  equal  parts.  ^.  $3.15c.  6/o%m. 

19.  Divide  150.15  barrels  into  32  equal  parts.       A.  4.6921875 

20.  At  $.5  a  yard,  how  many  yards  will  cost  $25 1 

.  5  )  2  5  ■  0  See  the  rule  (10)  for  annexing  ciphers; 

A.        5     0  yd.     which  makes  the  quotient  a  whole  number 

21.  Divide  30  hogsheads  by  .15  of  a  hogshead.  A.  200. 

22.  How  many  times  does  60.15  exceed  .0000051 

A.  12,030,000. 

23.  At  .5  of  a  dollar  or  50  cents  a  yard,  how  much  may  be  bought 
for  .35  of  a  dollar  or  35  cents "? 

.  5  )  .  3  5  Make  the  7  a  decimal,  then  the  decimals  in  .7 

A.     ~7  yd.      and  .5  will  equal  those  in  .35.  (Rule  9.) 


24.  Divide  .192816  by  .312.  A.  .618. 

25.  Divide  .25008  by  .48.  A.  .521. 

26.  What  is  the  quotient  of  .00025  divided  by  .25 1 

.25).  00025  Prefixing  2  ciphers  to  1  makes  the 

A (\  0  i         decimal  places  in  .001  and  .25  equal  to 

'      those  in  .00025.  (Rule  9.) 

Q.  Why  is  .75 -r- 5  =  .15?  12.  Why  is  .75 h- .25=^3?  15r~Wh7  isTo.75-f- 
8=10.09375?  17.  Why  is  .25-^.5  =  50?  20.  Why  is  .35 ^.5  =  .7?  23.  What 
is  the  quotient  of  10.8  divided  by  12?— 10.8  by  .12?— .108  by  1.2?— .108  by 
.12?— .108  by  12?— 1.08  by  1.2?— 1.08  by  .12?— 10.8  by  1.2?— 108  by  .12?— 
I08by  1.2?— l.OSby  12? 

\ 


160  ARITHMETIC. 

27.  Divide  37.035  dollars  into  12345  equal  parts.  A.  .003. 

28.  Divide  1.77975  into  25425  equal  parts.  A-  .00007. 

29.  Since  decimals  decrease  from  the  left  towards  the  right  in  a 
tenfold  proportion ;  therefore, — 

30.  To  divide  by  10,  100,  1,000,  &c. — Merely  remove  the  separa- 
trix  one  place  further  towards  the  left  for  every  cipher  in  the  divisor. 

31.  Divide  3752.3  by  10  ;— by  100  ;— by  1,000  ;— by  10,000  ;— by 
100,000;— by  1,000,000. 

A.  375.23;  37.523;  3.7523;  .37523;  .037523;  .0037523. 

32.  Divide  1561.275  by  24.3  ;— by  48.6  ;— by  12.15  ;— by  6.075. 

A.  64.25;  32.125;  128.5;  257. 

33.  Divide  8358  by  .6073.  A.  13762.55557.  + 

34.  Divide  .03315  by  .085.  A.  .39. 

35.  Divide  .264  by  .2  ;— by  .4  ;— by  .02  ;— by  .04  ;— by  .002  ;— by 
.004.  A.  1.32;   .66;  13.2;  6.6;  132;  66. 

36.  Suppose  a  bushel  of  corn  to  contain  15,000  grains,  and  to  cost 
$.90,  what  is  tlie  cost  of  a  single  grain?  A.  $.00006. 

37.  Divide  80  by  8  tenths ;— 800  by  8  hundredths. 

38.  Multiply  100  by  8  tenths  ; — 10,000  by  8  hundredths. 

39.  Divide  5,000  by  5  thousandths  ; — 5,000,000  by  5  millionths. 

40.  Multiply  1  million  by  .005  ; — 1  trillion  by  .000005. 

41.  How  many  times  greater  is  8  than  .16? 

42.  Multiply  50  by  16  hundredths  of  a  unit. 

43.  How  many  times  greater  is  $16  than  2  cents  or  $.02  ? 

44.  Multiply  800  by  2  hundredths  of  a  dollar. 

45.  How  many  times  are  5  cents  contained  in  5  dollars  ? 

46.  Multiply  100  by  5  hundredths  of  a  dollar. 

47.  Divide  $196.08  by  $5.16.  A.  38. 

48.  Divide  $2.15565  by  $1.05.  A.  2-053. 

49.  Wlien  the  quotient  repeats  one  figure  or  more,  continue  the 
division  for  three  decimal  places  ;  then  write  a  9  under  the  repeating 
figure  in  the  fourth  place,  and  the  quotient  will  express  the  exact 
decimal,  (lxix.  Case  ii.  17.) 

50.  Divide  $7  by  12  cents,     (f =f )  A.  58.333^. 

51.  Multiply  58.333i  by  12  hundredths  of  a  dollar. 

52.  What  is  the  price  of  cloth  by  the  yard  when  6  yards  cost  $20  ? 
when  9  yards  cost  $4  ]  A.  $3f  or  $3,333^;  $.444|. 

53.  Divide  272^  by  30j  decimally.  A.  9. 

54.  Divide  272|  by  9  decimally.  A.  30.25. 

55.  Divide  30|  by  5|  decimally.  A.  5.5. 

56.  How  many  times  greater  is  £6.  5s.  than  12s.  6d.?  Reduce 
both  to  the  decimal  of  a  pound  first.  A.  10  times. 

57.  Divide  £200.  16|s.  by  2s.  6d.  decimally.       A.  jei606.48. 

58.  Suppose  200A.  3]rd.  of  land  to  cost  £1,700.  3s.  4d.  3^qr.  ; 
what  is  the  price  per  acre  T  A.  £8.5=£8.  10s. 

59.  If  a  ship  in  sailing  from  New  York  to  Canton  occupy  207  days 


REDUCTION    OF    CURRENCIES. 


161 


16h.  26m.  and  24sec.,  how  many  such  trips  would  require  170Y. 
255d.  Uh.l  A.  300. 

60.  A  gentleman  having  returned  from  a  voyage  at  sea,  found  by 
actual  calculation  that  he  had  sailed  just  5,475  miles  6fur.  24rd., 
being  on  an  average  60  miles  6fur.  29rd.  3yd.  lOin.  and  2fb.c.  each 
day  during  the  voyage ;  how  many  days  was  he  in  performing  the 


voyage 


A.  90  days. 


REDUCTION    OF    CURRENCIES. 

LXXIV.  1.  Formerly,  all  accounts  in  the  United  States  were 
kept  in  the  currency  of  Great  Britain,  that  is,  in  pounds,  shillings, 
pence,  and  farthings,  called  sterling  money,  which  was  at  first  oJ 
uniform  value  in  both  countries.     See  vii.  table  2. 

2.  But  previous  to  the  adoption  of  federal  money,  the  pound,  and 
consequently  its  divisions,  had,  under  the  same  names,  received  dif 
ferent  values  in  different  sections  of  the  union,  as  in  the  following 

3.    TABLES    OF    CURRENCIES. 


NEW    ENGLAND    STATES,  VIRGINIA, 
KENTUCKY  AND  TENNESSEE. 

1  pound^  =$3,335=  $V" 

6  shillings  =S1.00  =£t\ 

9  pence  =$  .12^=  $1 

4^  pence  =$  .06}=  SyV 

N.  YORK,  N.  CAROLINA  AND  OHIO. 


N.  JERSEY,    PENNSYLVANIA,   DELA 
WARE,  MARYLAND  AND  LOUISIANA. 

1  pound^  =$2.66|=$f 

7s.  6  pence  =$1.00  =£  | 

11|  pence  =$  .12|=  $^ 

5d.  2|qr.  =$  .06|=  $Jg 

SOUTH  CAROLINA  AND  GEORGIA. 


1  pound^ 

=$2.50  =  sy* 

1  pound* 

=$4,284=$  t" 

8  shillings 

=$1.00  -£j\ 

4s.  8  pence 

=$1.00    =£-^j; 

1  shilling 

=S  .12^=  $i 

7  pence 

=$  .121=$    I 

6  pence 

=$  .06^=$3V 

3|  pence 

=$  .001=$^ 

IN  CANADA  AND  NOVA  SCOTIA  CURRENCY. * 

5  shi]lings=$1.00=jei-,  and  £l=$4:. 

IN    ENGLI.3H    OR    STERLING    MONEY.* 

4s.6d.=  $1.00=jg/o,  and  £l=$4.44§=$  V- 

LXXIV.  Q.  In  what  currency  are  our  accounts  kept  ?  2.  How  were  they 
formerly  kept?  1.  When  was  federal  money  established?  xxxi.  1.  What  is 
said  of  the  vahie  of  the  pound  in  the  different  states?  1,  2.  How  many  shil- 
lings make  a  dollar  in  the  different  states,  Canada,  and  Great  Britain?  See  3. 
Where  do  12l  cents  pass  for  1  shilling?— for  9  pence?— for  111  pence?— for  7 
pence?  What  does  6^  cents  pass  for  in  the  different  states?  What  part  of  a 
pound  is  $1  in  Maine,  N.  Carolina,  Delaware,  Georgia,  Canada,  and  sterling 
money?  See  tables,  3. 


162  ARITHMETIC. 

GENERAL  RULE  FOR  THE  REDUCTION  OF  CURRENCIES. 

4.  Multiplying  the  pounds,  icilh  the  shillings,  pence,  <5fc.  reduced 
to  the  decimal  of  a  pound,  hy  the  fraction  that  expresses  ivhat  part 
£\  is  0/  $1,  found  in  the  tables,  produces  federal  money;  and 
reversing  the  process  produces  pounds  again. 

6.  Or,  which  is  the  same  thing — Annexing  a  cipher  to  any  sum  of 
pounds  in  Neio  England  or  New  York  currency,  and  dividing  by  half 
the  number  of  shillings  in  a  dollar,  produces  federal  money. 

6.  Change  X^QOO  N.  E.  currency  to  dollars.    (JCI  =$  V  x  900.) 

7.  Change  $3,000  to  N.  E.  currency.     ($1=  £j%  x  $3,000.) 

8.  Change  jC270  N.  E.  currency  to  federal  money. 

9.  Change  900  dollars  to  N.  E.  currency. 

10.  Change  £150  Pennsylvania  currency  to  dollars. 

11.  Change  400  dollars  to  the  currency  of  Pennsylvania. 

12.  Change  £3.  8s.  3d.  S.  Carolina  currency  to  federal  money. 
Note.— i;3.8s.3d.=: £3.4125  x  ^^^  =  $14.6250,  A.  See  lxix.  c  rv 

13.  Change  $14,625  to  S.  Carolina  currency. 

Note.— $14,625  x  ^  =£3.4125=  je3.8s.3d.  A.  See  lxix.  case  v. 

14.  Change  £17.  15s.  6d.  N.  E.  currency  to  dollars. 

15.  Change  $59.25  to  New  England  currency. 

16.  Change  £8.  15s.  6d.  New  York  currency  to  dollars. 

17.  Change  $21.9375  to  N.  Carolina  currency. 

18.  Change  £12.  7s.  6d.  Virginia  currency  to  dollars. 

19.  Change  $41.25  to  Tennessee  currency. 

20.  Change  £5.  15s.  6d.  Delaware  currency  to  dollars. 

21.  Change  $15.40  to  Maryland  currency. 

22.  Change  £2.  2s.  Georgia  currency  to  dollars. 

23.  Change  $9  to  South  Carolina  currency. 

24.  Change  £252  Canada  currency  to  dollars. 

25.  Change  $1,008  to  Nova  Scotia  currency. 

26.  Change  £252  English  currency  to  federal  money. 

27.  Change  $1,120  to  English  or  sterhng  money. 

28.  A  merchant  in  Philadelphia  purchased  for  Messrs.  Robinson  & 
Pratt,  of  Quebec,  300  bales  of  cotton,  which  averaged  275  pounds 
per  bale,  at  a  cost  of  10^  cents  per  pound.  Now,  how  many  pounds, 
in  Canada  currency,  must  Robinson  &  Pratt  remit  to  their  agent  for 
the  payment  of  said  cotton  ? A.  £2,114.  Is.  3d. 

(29.)  London,  Jan,  4.  1840. 

Messrs.  Rice  &  Donaldson,  of  Philadelphia,  U.  S. 

Bought  of  James  Wellington. 
462  yds.  Blue  Broadcloth,  a  17s.  6 ^d. 
418  do.   Black       do.  a  14s.  9^d. 

519  do.   Black  Silk  a  9s.  3d.  

£954.  16s.  1\A. 
What  sum  in  federal  money  will  settle  the  above  bill  ? 

A.  $4243.6944.4- 


Q.  What  is  the  rule  for  reduction  of  currencies  ?  4,  5. 


RATE    PER    CENT.  163 


RATE    PER    CENT. 

LXXV.  1.  Per  Cent.,  from  the  Latin  yer,  by,  and  centum,  one 
hundred,  signifies,  hy  the  hundred. 

2.  Rate  Per  Cent.,  thavei'ore,  signiCies,  the  rate  by  the  hundred, 
that  is,  the  number  of  hundredths  of  any  sum  which  is  considered  as 
advanced,  expended,  gained,  or  lost. 

3.  Since  hundredths  are  decimals  of  two  places,  all  calculations  by 
the  rate  per  cent,  fall  very  properly  within  the  province  of  decimals. 

4.  Thus  5  per  cent,  of  any  sum  is  y-j^  of  that  sum ;  6  per  cent,  is 
rffo=-06;  15  per  cent,  is  yVV=15,  &c. 

5.  Then  5  percent,  of  $20  is  .05=yfo  of  $20=$l,and  6  per 
cent,  of  .$20  is  .06  or  y^^  of  $20=  $1.20. 

6.  Hence,  to  express  any  given  rate  decimally — When  it  consists 
of  one  figure  only,  ivrite  a  cipher  and  the  separatrix  before  it;  ivhe.n 
of  two  figures,  only  the  separatrix ;  when  of  three  or  more  figures, 
call  only  the  two  right  hand  ones  decimals. 

7.  Thus  1  per  cent,  is  .01  ;  10  per  cent,  is  .10=.l ;  23  per  cent, 
is  .23;  31.5  per  cent,  is  3.15. 

CASE   I. 

To  fiind  the  value  of  the  rate  per  cent. 

RULE. 

8.  Either  tnultiply  by  the  rate  per  cent,  as  a  decimal,  or  as  a  whole 
number ;  but  in  the  latter  case  point  off  ttvo  figures  in  the  product, 
{for  dividing  by  100.) 

9.  Suppose  a  trader  has  $1,500,  and  wishes  to  lay  out  6  per 
cent,  of  it  in  silks,  how  many  dollars  will  the  silk  cost  1 

$1500        Or  $1500  For  6  per  cent.=  .06  or 

.  0  6 6        yf^.     To  divide  by  100,  cut 

^.$90.00      .A.  $90.00        off  two  figures  on  the  right. 


10.  What  is  5  per  cent,  of  $8,000]  A.  $400. 

11.  What  is  25  per  cent,  of  $400  1  A.  $100. 

12.  What  is  50  per  cent,  of  $3001  A.  $150. 

13.  What  is  75  per  cent,  of  $6001  A.  $450. 

14.  What  is  100  per  cent,  of  $401  A.  $40. 

15.  Hence,  tvhen  the  rate  is  100  per  cent,  the  given  sum  itself  is 
the  value  of  the  given  rate ;  when  75  per  cent,  the  value  is  J  of  the 
given  sum ;  ivhen  50  per  cent,  it  is  ^  ;  ivhen  25  per  cent.  \  ;  and  so  on. 

16.  Suppose  a  man  who  has  $5,000  pays  away  25  per  cent,  of  it, 

LXXV.  Q.  What  is  the  meaning  of  per  cent.?  1 .  What  of  rate  per  cent,?  2. 
How  is  the  rate  per  cent,  determined  ?  3.  What  is  meant  by  5  or  6  per  cent.?  4. 
What  is  5  per  cent,  of  $20?— of  $30  ?— 6  per  cent,  of  S20  ?— of  $30?  How  is 
any  per  cent,  expressed  decimally  ?  6.  What  is  the  decimal  expression  for  1 
per  cent.? — for  9  per  cent.? — for  10  per  cent.? — for  23  per  cent.? — for  315  per 
cent.?  What  is  the  rule  for  finding  the  value  of  the  rate  per  cent.?  8.  What  is 
1  per  cent,  of  $2,000?— 2  per  cent,  of  $4,000?— C  per  cent,  of  $10,000? 


164  ARITHMETIC. 

how  many  dollars  has  he  left  1     The  whole  is  100  per  cent.;  25  per 
cent,  deducted,  leaves  75  per  cent,  or  f  of  the  whole.    A.  $3,750, 

17.  Suppose  a  man  having  an  estate  worth  in  cash  $1,000,  spends 
15  per  cent,  of  it  in  one  year,  35  per  cent,  the  second  year,  and  the 
remainder  in  the  third  year ;  what  per  cent,  does  he  spend  the  last 
year,  and  how  many  dollars  is  it]  A.  50  per  cent.;  $500. 

18.  Suppose  a  flour  merchant  sells,  in  the  course  of  the  year, 
8,000  barrels  of  flour,  at  an  average  price  of  $9 ^  a  barrel,  and  depo- 
sits 20 1  per  cent,  of  the  money  in  the  bank  ;  how  many  dollars  does 
he  deposit?  A-  $10,120. 

19.  What  is  6  per  cent,  of  $40.27  \  A.  $2.4102. 

20.  What  is  15  per  cent,  of  $62.50?  A.  $9.37|. 

21.  What  is  20f  per  cent,  of  25.561b.?  A.  5.28241b. 

22.  What  is  5f  per  cent,  of  $65,375?  A.  $3,759.+ 

23.  Suppose  a  planter  grows  40,000  pounds  of  cotton  in  a  year, 
and  sells  30}  per  cent,  of  it  for  15  cents  a  pound  ;  44^  per  cent,  of  it 
for  16 g  cents  per  pound;  what  will  his  sales  amount  to,  and  how 
much  of  his  crop  remains  on  his  hands  ?  A.  $4,701.37^- ;  10,0001b. 

24.  What  is  I  per  cent,  of  $200?  "        A^.15. 

25.  What  is  /„  per  cent,  of  $53,625  ?  A.  $.321  J. 

26.  When  the  per  centage  is  very  small,  it  is  sometimes  expressed 
in  cents;  thus  30  cents  of  $100  is  100  times  smaller  than  30  per 
cent.,  which  is  $30  of  $100  ;  therefore, — 

27.  When  the  rate  per  cent,  is  expressed  by  cents — Consider  two 
more  places  in  the  product  decimals  on  that  account. 

28.  What  is  the  difference  between  20  per  cent,  of  $500,  and  20 
cents  for  every  hundred  dollars  of  $500  ?  A.  $99. 

20.  When  the  rate  per  cent,  is  10  cents  of  $100,  what  is  its  value 
on  $1,000,000?  $1,000. 

30.  When  the  rate  per  cent,  is  5  cents  of  $100,  what  is  its  value  on 
$6,834.5625?  A.  f3.417.+ 

CASE    II, 
When  the  given  sum  is  a  compound  quantity. 
RULE. 

31.  Reduce  it  first  to  a  decunal  of  the  highest  denomijiation  men- 
tioned, then  multiply  hy  the  rate  as  before ;  after  which  find  the  value 
of  the  decimal  in  a  compound  quantity  again,   [lxix.  cases  iv.  and  v,] 

Q.  When  the  rate  is  100  per  cent,  what  will  always  be  its  value  ?  15.  What 
■when  75  per  cent.?  15,  When  50  per  cent.?  15,  When  25  per  cent.?  15, 
What  then  is  the  value  of  100  per  cent,  of  $300  ?— 75  per  cent,  of  400  guineas? — 
50  per  cent,  of  144  eagles? — 25  per  cent,  of  800  gallons ?  What  is  the  value  of 
25  per  cent.,  10  per  cent.,  and  20  per  cent.,  of  $1,000?  Suppose  a  man  has 
$400,  and  loses  10  per  cent,  of  it,  how  many  dollars  has  he  left?  What  is  ll 
per  cent,  of  $200?— of  $500?  What  is  5  per  cent,  of  $120?  What  is  1  per 
cent,  of  $120? — of  $240?  When  the  per  centage  is  very  small,  how  is  it  fre- 
quently expressed?  26,  What  is  the  direction  in  such  cases?  27.  When  the 
§er  centage  is  30  cents  for  every  $100,  what  does  it  amount  to  on  $2,000? — on 
115,000?  When  the  given  sum  consists  of  several  denominations,  how  do  you 
proceed?  31. 


RATE    PER    CENT.  185 

32.  What  is  5  per  cent,  of  jGTS.  9s.  6d.1    (=£75.475.) 

A.  £3.  15s.  5M.+ 

33.  What  is  5  per  cent,  of  £400.  17s.  3d.?     A.  £20.  lOd.  Ifqr. 

34.  What  is  101^  per  cent,  of  2cwt.  3qr.  71b.? 

A.  2cwt.  3qr.  lllb.+ 

35.  What  is  10  per  cent,  of  5001.  6fur.  30rd.?        A.  501.  27pd. 

CASE   III. 

,  To  find  the  rate  from  having  its  value  given. 

36.  What  per  cent,  is  $30  of  $500  ?  Since  6  per  cent,  of  $500 
is  found  by  case  i,  thus,  $500  x.06=  $30  :  then  $30  ^$500  must  re- 
produce the  rate. 

RULE. 

37.  Of  the  two  given  sums,  divide  the  one  expressing  the  value  of 
the  rate  hy  the  other. 

38.  What  per  cent,  of  $800  is  $56  ?  A.  7  per  cent. 

39.  What  per  cent,  of  $700  is  $63  ?  A.  9  per  cent. 

40.  What  per  cent,  of  $200  is  $30  ?  A.  15  per  cent. 

41.  If  a  man  has  300  bushels  of  rye,  and  sells  60  bushels,  what 
per  cent,  of  the  whole  does  he  sell?  A.  20  per  cent. 

42.  Recollect  that  the  rate  per  cent,  is  always  equal  to  so  many 
hundredths ;  thus  .06  is  yfg- ;  therefore,  it  is  6  per  cent.;  .09  is  y^ 
=  9  per  cent.;  .12=y/o=  12  per  cent. 

43.  Then,  as  the  rate  per  cent,  is  restricted  to  two  decimal  places, 
call  the  other  figures  on  the  right  of  hundredths  in  the  quotient  deci- 
mals of  hundredths,  or  of  the  rate  per  cent. 

44.  Thus,  .062  is  6^^=64  per  cent;  .1525  is  lbj%%=\!j\  per 
cent.;  .27375=  27y^^7  5_=:27i  per  cent. 

45.  What  per  cent,  of  $20  is  $1.70  ?  A.  .085=  81 

46.  What  per  cent,  of  $200  is  $6.40  ?  A.  .032=  3|. 

47.  What  per  cent,  of  $300  is  $13.20?  A.  .044=  4|. 

48.  What  per  cent,  of  $60.25  is  $4.6995?  A.  7|. 

49.  What  per  cent,  of  $200  is  $1  ?  $1  ^200=  .005,  and  as  there 
are  no  hundredths,  the  5  is  x^o=i  P^r  cent.  A.  \  per  cent. 

60.  What  per  cent,  is  45  cents  of  S60  ?  A.  .0075=yVo  =£• 

51.  Suppose  a  lawyer  charges  $5  for  collecting  a  debt  of  §4,000, 
what  per  cent,  is  it  ?  A.  .00125=yWo  =1- 

52.  Suppose  a  grocer  finds  that  in  retailing  2,500  gallons  of  mo- 
lasses, it  does  not  hold  out  by  10  gallons,  what  per  cent,  of  the  whole 
is  the  loss  ?  A.  |  per  cent. 

CASE   IV 

To  find  any  sum  from  having  its  rate  per  cent,  and  its  value  given. 

Q.  What  is  the  rule  for  finding  the  rate  ?  37.  Why?  36.  What  per  cent,  is 
$5  on  $20?— S4  on  .$24  ?— $6  on  $150?  How  is  the  per  centage  expressed 
decimally  ?  42.  When  it  is  expressed  by  three  or  more  decimal  figures,  what 
per  cent,  do  all  the  figures  except  the  two  left  hand  ones  express?  43.  What 
per  cent,  then  is  .062  ?— is  .1525?  44.  Of  what  sum  is  $5  only  2  per  cent..? 
What  is  the  rule  for  it?  54.  Of  what  sum  is  2  per  cent.  $10  ?  Of  what  sum  is 
8  per  cent.  $10? 


166  ARITHMETIC. 

53.  A  man  failing  in  business  is  able  to  pay  S720,  which  is  only  6 
per  cent,  of  what  he  owed  ;  how  much,  then,  does  he  owe  ? 

•  ^  ^  )$  720  .  00        Since  812,000  x  .06=720,  therefore  $720 
A.     $12,000     -^.OGmust  reproduce  the  $12,000. 

RULE. 

54.  Divide  the  value  of  the  rats  per  cent,  hy  the  rate  itself  expressed 
as  a  decimal. 

55.  If  a  man  owes  a  note  at  bank,  and  pays  12  per  cent,  of  it, 
making  $300,  what  was  the  face  of  the  note  ?  A.  $2,500. 

56.  Of  what  sum  is  $50  only  4  per  cent.]  A.  $1,250. 

57.  Of  what  sum  is  $50;^  per  cent.?     (  =  .005.)        A.  $10,000. 

58.  Suppose  I  paid  $750,  which  Vas  f  per  cent.,  for  the  collection 
of  a  certam  debt ;  how  large  was  the  debt?  A.  $100,000. 


STOCKS. 

LXXVI.  1 .  Stocks  is  a  general  name  for  all  funds '  invested  in 
banks  and  other  corporate  ^  bodies. 

2.  Stocks  consist  of  shares  ^  of  an  equal  amount,  as  $50,  or  $100 
each,  but  usually  of  $100,  and  as  such  are  bought  and  sold. 

3.  The  NOMINAL  VALUE  of  a  share  is  what  it  costs  at  first,  and  when 
it  sells  for  that  in  the  market,  it  is  said  to  be  at  par.^  The  nominal 
value,  then,  is  its  par  value. 

4.  The  REAL  VALUE  of  a  share  is  what  it  actually  sells  for,  which 
often  varies  at  different  times. 

5.  When  stock  sells  for  more  than  its  par  value,  it  is  said  to  be 
above  par,  or  at  an  advance ;  ®  but  when  for  less,  below  par,  or  at  a 
discount.'' 

RULE. 

6.  The  various  values  of  stock  being  estimated  at  so  much  per 
cent.,  they  are  calculated  as  in  the  preceding  article. 

LXXVI.  Q.  What  is  understood  by  stocks  ?  1.  Of  what  do  they  consist  ?  2. 
What  is  meant  by  their  nominal  value  ?  3.  What  by  their  real  value  ?  4. 
What  names  are  used  to  designate  the  market  price  of  stocks  ?  5.  What  is  the 
rule  for  ascertaining  the  value  of  stocks  ?  6.  What  is  the  value  of  $500  stock 
at  8  per  cent,  advance  ? — at  5  per  cent,  discount  ? — at  2  per  cent,  above  par  ? — 
at  10  per  cent,  below  par  ? — at  par  ? 

1  Funds.  Capital ;  a  sum  of  money  appropriated  for  commercial  or  other  operations. 
A  stock  or  capital  intended  to  furnish  supplies  of  any  kind.  Money  lent  to  government. 
Abundance;  ample  stock  or  store. 

2  Corporate.  United  in  one  body  or  community,  as  a  number  of  persons  who  are 
allowed  by  law  to  sue  and  be  sued,  <5 c,  as  if  they  were  but  one  person.  United ;  collec- 
tively one. 

3  Share.  Apart;  a  portion;  a  quantity;  dividend;  a  part  contributed.  The  broad 
iron  or  blade  of  a  plough.     To  go  shares,  to  be  equally  interested. 

4  Nominal.  Not  real;  existing  only  in  name. 

5  Par.  The  Latin  for  equal. 

6  Advance,  ftlovement  forward,  or  a  gradual  progressing  in  any  thing.  Promotion  j 
preferment ;  first  step  towards  agreement.  A  furnishing  of  goods  or  money  for  others 
In  advance,  in  front ;  before. 

7  Discount.  A  sum  deducted ;  deduction  for  prompt  pay. 


COMMISSION.  167 

7.  What  is  the  value  of  $3,500  of  stock  at  8  per  cent,  advance  ? 
100+8  =  108  per  cent.  X  $2,500=82,700.  A. 

8.  A  gentleman  purchased  35  shares,  of  $100  each,  in  the  United 
States  Bank,  at  8f  per  cent,  advance  ;  what  did  they  cost  1 

A.  $3801.875. 

9.  What  is  the  value  of  $4,500  stock  at  a  discount  of  5]  per  cent.? 

A.  $1,263.75. 

10.  A  merchant  subscribed  for  25  shares,  of  $50  each,  in  a  certain 
railroad,  which  declined  afterwards  15^  per  cent,;  what  was  its  real 
value  then.  A.  $1,053,125. 

11.  What  is  the  difference  between  $2,200  stock  at  par  value,  and 
at  an  advance  of  7yY  per  cent.?  A.  $1G0. 

12.  Suppose  I  purchased  stock,  the  par  value  of  which  is  $20,000, 
for  2  per  cent,  discount,  and  sold  it  for  2  per  cent,  advance ;  how 
much  did  I  make  on  it  1  A.  $800. 

13.  Suppose  an  original  subscriber  for  100  shares,  of  $50  each,  in 
the  Exchange  Bank,  Providence,  receives  a  dividend  of  $100  ;  what 
per  cent,  is  that  on  his  stock  1  [See  lxxv.  case  iii.]  A.  2  per  cent. 

14.  What  per  cent,  is  equal  to  a  dividend  of  $1,000  on  $20,000 
stock?  A.  5  per  cent. 

15.  Suppose  you  receive  a  dividend  of  $1,000,  being  at  the  rate  of 
5  per  cent.,  on  your  bank  stock;  what  amount  of  stock  have  you? 
[See  LXXV.  case  iv.]  A.  $20,000. 

16.  Suppose  a  merchant  to  have  been  an  original  subscriber  for 
500  shares,  of  $50  each,  in  the  Bank  of  America,  payable  by  install- 
ments, as  follows  : — ^  in  three  months,  which  he  sold  for  5}  per  cent, 
advance ;  f  in  six  months,  which  brought  him  6|  per  cent,  advance  ; 
and  the  balance  in  nine  months,  which  he  was  compelled  to  sell  at 
8f  per  cent,  discmnt ;  what  did  he  gain  by  the  whole  transaction  ? 

^  A.  $808,333.+ 


COMMISSION. 

LXXVII.  1.  Commission  is  an  allowance  of  so  much  per  cent, 
made  to  factors,  brokers,  and  other  agents,  for  their  services  in  buy- 
ing and  selling  for  their  employers. 

2.  A  Factor  is  a  person  employed  by  another  at  a  distance,  to 
transact  business  on  his  account. 

3.  A  Broker  is  a  person  who  deals  in  stocks,  goods,  &c.,  or  ex- 
changes money,  either  on  his  own  account  or  for  others. 

LXXVII.  Q.  What  is  Commission?  1.  Factor?  2.  Broker?  3.  What  is 
the  rule  ?  4.  How  many  dollars  does  the  commission  of  2  per  cent,  on  $300 
amount  to  ? — 5  per  cent,  on  $400?  How  many  dollars  must  I  pay  for  changing 
$5,000,  when  the  brokerage  is  ^  per  cent.?  How  much  for  changing  $2,000  at 
the  same  rate  ? 


168  ARITHMETIC. 

RULE. 

4.  The  rule  is  the  same  as  that  for  estimating  the  value  of  stocks. 

5.  If  my  agent  sells  goods  amounting  to  $400,  what  is  his  commis- 
sion at  2|  per  cent.'?     $400  x  2^  per  cent.  =$10.  A. 

6.  My  correspondent  writes  me  that  he  has  purchased  goods  to 
the  amount  of  85,000  ;  what  will  his  commission  amount  to  at  3  per 
cent.1  A.  $150. 

7.  What  is  the  commission  on  $417  at  1  per  cent.l — at  1^  per 
cent.1 — at  2  per  cent.? — at  3^  per  cent.] — at  4f  per  cent.? 

A.  $4.17;  $6.25i;  $8.34;  $14.59^;  $19.8075. 

8.  Suppose  a  factor  purchases  300  pounds  of  indigo  for  $2.50  a 
pound  ;  what  will  his  commission  amount  to  at  6^^  per  cent."? 

A.  $48.75. 

9.  What  does  a  broker  exact  for  changing  $3,700  in  bills  on  the 
Bank  of  America,  New  York,  for  the  same  sum  on  the  Phoenix  Bank, 
Connecticut,  the  rate  of  exchange  being  |  per  cent.]    A.  $13.87^. 

10.  Suppose  a  broker  purchase  on  your  account  $20,000  stock  in 
the  "Great  Western  Canal,"  at  13 J  per  cent,  advance;  what  will 
the  stock  cost,  and  what  will  his  brokerage  amount  to  at  the  rate  of 
I  per  cent.'?  A.  $22,625,  cost ;  $28.28^. 

11.  If  a  broker  receive  80  cents  for  changing  $400,  at  what  per 
cent,  does  he  reckon  the  exchange "?  ^-  i  per  cent. 

12.  Suppose  I  remit  to  my  factor  in  New  York,  $6,000,  for  the 
purchase  of  flour,  and  he  writes  me  that  the  flour  cost  $10  a  barrel, 
and  that  his  commission  is  2h  per  cent.;  how  many  barrels  of  flour 
shall  I  receive,  after  deducting  his  commission '?     A.  585  barrels. 

13.  Suppose  a  commission  merchant  sells  on  my  account  200;^ 
bales  of  cotton,  each  bale  containing  3151b.  8oz.,  at  13  cents  a  pound, 
for  a  note  at  six  months,  and  charges  2|  per  cent,  for  selling,  and  3 
per  cent,  more  for  guaranteeing  the  paper ;  what  will  be  the  balance 
due  me  1  A.  $7759.587.+ 

14.  A  commission  merchant  sold  200hhd.  7gal.  3qt.  1.504pt.  of 
molasses  for  $30  a  hogshead,  for  which  he  charged  3  per  cent,  com- 
mission ;  what  will  be  the  balance  due  his  employer  1 

A.  $5823.666.+ 


INSURANCE. 

LXXVIII.  1.  Insurance  is  a  security,  by  paying  a  stipulated 
sum,  called  a  premium,  to  indemnify  the  party  insured  against  such 
losses  on  ships,  houses,  goods,  &c.,as  may  happen  from  storms,  fire, 
or  other  accidents. 

LXXVIII.  What  is  Insurance?  1.  A  policy?  2.  How  is  the  cost  of  in- 
suring estimated  ?  2.  How  many  dollars  must  you  pay  for  procuring  an  insu- 
rance on  your  house,  valued  in  the  policy  at  $10,000,  the  rate  being  |  per  cent.'' 


LOSS    AND    GAIN.  169 

2.  The  Contract  of  Indemnity^  is  called  a  Policy,  and  the  pre- 
mium paid  for  it  is  usually  stated  at  so  much  per  cent,  as  in  the  fore- 
going articles. 

3.  What  is  the  premium  for  insuring  an  East  India  ship  valued  at 
S25,000,  at  15^  per  cent.  1  A.  3,875. 

4.  What  is  the  premium  for  insuring  S2,000  at  2^  per  cent.  1 — 3 
per  cent.  1 — 4|  per  cent. — 5|  per  cent.  1 — 7-^^  per  cent. — 8y^  per 
cent.?  A.  $50;  $60;  $87.50;  $105;  $143;  $165.33^. 

5.  If  you  effect  an  insurance  on  your  house  for  $5,000  at  ^  per 
cent,  per  annum,  what  would  it  amount  to  in  5  years  1    A.  $93.75. 

6.  A  manufacturer  effected  an  insurance  on  his  factory  building 
and  machinery,  both  valued  in  his  policy  at  $15,600,  paying  a  premi- 
um of  2  per  cent,  per  annum.  In  the  second  year  the  establishment 
suffered  by  fire,  as  was  estimated,  $1,200  ;  how  much  did  he  save  by 
the  insurance  1  A.  $576. 

7.  Suppose  you  pay  a  premium  of  30  cents  on  $100  for  insuring 
$10,000  on  your  house,  $2,000  on  your  furniture,  $450  on  your 
books,  $600  on  your  span  of  horses,  and  $350  on  your  harnesses, 
buffalo  robes  and  saddles  ;  how  many  dollars  does  the  insurance  on 
all  these  articles  cost  1  ^.$40.20. 


LOSS  AND  GAIN. 

CASE   I. 

LXXIX.  To  find  the  sum  gained  or  lost. 

RULE. 

1.  Find  the  value  of  the  given  rate  per  cent,  as  before. 

2.  If  I  buy  goods  amounting  to  $1,675,  and  sell  them  for  15  per 
cent,  gain,  what  are  my  profits'?  A.  $251.25. 

3.  If  I  sell  goods  for  10  per  cent,  loss,  which  cost  m.e  $500,  how 
many  dollars  do  I  lose  by  them  1  A.  $50. 

4.  Suppose  I  buy  400|  barrels  of  flour  for  $16f  a  barrel,  and  sell 
it  for  I  per  cent,  advance  ;  how  many  dollars  do  I  gain  by  it  \ 

A.  $25,13|.+ 
CASE  II. 

To  find  what  price  must  be  demanded  to  gain  or  lose  a  certain  per 
cent. 

RULE. 

5.  If  the  given  rate  is  gain  per  cent,  add  it  to  100  and  multiply  the 

LXXIX.  Q.  When  goods  worth  $500  are  sold  for  10  per  cent,  loss,  what  do 
they  bring?  Wliat  is  the  rule?  1.  When  calico  costs  30  cents  per  yard,  at 
what  price  must  it  be  sold  to  gain  5  per  cent.? — to  lose  6  per  cent.? — to  gain  10 
per  cent.?  What  is  the  rule  ?  5.  How  is  the  gain  or  loss  per  cent,  ascertained  ?  13. 

1  Indemnity.  Security  given  to  save  from  harm  or  loss  ;  security  against  puaisli  • 
meat. 

15 


170  ARITHMETIC. 

cost  hy  that  sum ;  but  if  the  rate  is  loss  per  cent,  deduct  it  from  100 
and  multiply  hy  the  remainder. 

0.  If  I  buy  a  quantity  of  wheat  for  $200  and  wish  to  gain  10  per 
cent,  by  the  sale  of  it,  what  must  I  ask  for  it  1  A.  ^220. 

7.  A  man  paid  $50  for  goods  which  he  purchased  at  auction,  and 
which  he  was  glad  to  sell  at  a  loss  of  10  per  cent. ;  what  did  he  re- 
ceive for  them  ?  A.  $45. 

8.  A  man  bought  a  cow  for  $44  and  sold  it  for  15  per  cent,  ad- 
vance ;  what  did  he  get  for  it  ?  A.  $50.00. 

9.  A  merchant  bought  a  hogshead  of  molasses  for  $44,  and  by 
accident,  9  gallons  leaked  out ;  what  must  he  sell  the  remainder  for 
per  gallon,  so  as  not  to  lose  but  10  per  cent.  1  A.  73j  cents. 

10.  A  merchant  bought  1 08  barrels  of  flour  for  $  lOj-  a  barrel ;  paid 
for  carting,  6  cents  a  barrel,  and  for  assistance  in  storing  it  $1,50 ; 
now  how  much  must  he  ask  a  barrel  for  it  to  gain  20  per  cent.  1 

A.  $12,388+. 

1 1 .  A  wholesale  dealer  in  flour  bought  in  one  week  the  following 
lots,  viz. — 118  barrels  for  $9^  per  barrel,  212  barrels  for  89^  per 
barrel,  315  barrels  for  $9  J  per  barrel,  and  400  barrels  for  $10  per 
barrel.  His  store  rent  was  $12.50  a  week;  clerk  hire,  $17  per 
week,  and  insurance  gV  per  cent. ;  what  price  per  barrel  will  cover 
all  the  expenses  and  afford  him  15  per  cent,  profit  1  A.  $11,018.+ 

CASE    III. 

To  find  the  gain  or  loss  per  cent. 

RULE. 

12.  First  find  what  the  gain  or  loss  is  hy  subtraction,  then  divide 
the  remainder  by  the  actual  cost. 

13.  A  merchant  bought  molasses  for  24  cents  a  gallon,  which  he 
sold  for  30  cents  ;  what  was  his  gain  per  cent.  ? 

A.  .25=25  per  cent. 

14.  A  grocer  bought  a  hogshead  of  wine  for  $75,  from  which  sev- 
eral gallons  leaked  out ;  the  remainder  he  sold  for  $60.  What  did 
he  lose  per  cent.  ]     [.02  is  2  per  cent.,  but  .2  is  20  percent.] 

A.  20  per  cent. 

15.  A  man  bought  a  piece  of  cloth  for  $20,  and  sold  it  for  $25 ; 
what  did  he  gain  per  cent. '?  A.  25  per  cent. 

16.  A  grocer  bought  a  barrel  of  flour  for  $8,  and  sold  it  for  $9  ; 
what  was  the  gain  per  cent.  ?  A.   12|  per  cent. 

17.  A  merchant  bought  a  quantity  of  goods  for  $318.50,  and  sold 
them  for  $299.39  ;  what  was  his  loss  per  cent.  1      A.  6  per  cent. 

18.  If  I  buy  a  horse  for  $150,  and  a  chaise  for  $250,  and  sell  the 
chaise  for  $350  and  the  horse  for  100,  what  is  my  gain  per  cent.  1 

A.   12|  per  cent. 

19.  Bought  20  barrels  of  rice  for  $20  a  barrel,  and  paid  for  trans- 
portation of  it  50  cents  a  barrel ;  what  will  be  my  gain  per  cent,  in 
selling  it  for  $25.62-i^  a  barrel  1  A.  25  per  per  cent. 


INTEREST.  171 


INTEREST. 

LXXX.  1.  Interest^  is  the  premium^  given  by  the  borrower  to 
the  lender  for  the  use  of  money. 

2.  Interest  is  usually  reckoned  at  so  much  per  cent,  like  stocks ; 
it  however  differs  from  them  in  limiting  the  per  cent,  to  definite^  pe- 
riods* of  time. 

3.  Thus,  6  per  cent,  per  annum'^  means  $6,  and  8  per  cent,  per 
annum,  $8 ;  each  for  the  use  of  $100  for  one  year,  and  in  the  same 
proportion  for  a  longer  or  shorter  period  of  time. 

4.  The  rate  per  cent,  in  most  countries  is  established  by  law,  and 
if  a  higher  rate  be  exacted  it  is  called  usury.^ 

5.  The  legaF  rate  varies  in  different  countries  ;  thus,  it  is  5  per 
cent,  in  England,  and  C  per  cent,  in  our  country  with  the  following 
exceptions. 

6.  In  New  York,  S.  Carolina,  Michigan  and  Wisconsin,  it  is  7  per 
cent. ;  in  Georgia,  Alabama  and  Mississippi,  8  per  cent.  ;  and  in 
Louisiana,  5  per  cent.* 

7.  When  no  mention  is  made  of  the  rale  of  interest,  the  lawful  one 
is  always  understood. 

8.  If  A  borrows  of  B  SI, 000,  and  agrees  to  pay  him  6  per  cent,  inter- 
est, that  is,  $6  for  every  $100,  making  S60  interest,  A  evidently  owes 
B  at  the  year's  end,  $1000  and  -SOO,  making  $1060. 

9.  The  sum  lent  is  called  the  Principal  ;  the  per  cent,  agreed  on, 
the  Rate  ;  and  the  principal  and  interest  added  together  the  Amount 

CASE  I. 

To  find  the  interest  for  one  year.     [See  LXXV.  8.1 

RULE. 
1.  Multiply  the  principal  hy  the  given  rate,  and  cut  off  two  more 

LXXX.  Q.  What  is  Interest?  1.  How  is  it  reckoned?  2.  In  what  respect 
does  it  differ  from  per  cent.?  2.  What  does  6  per  cent,  per  annum  mean  ?  3. 
What  is  usury?  4.  What  legal  rates  are  mentioned?  See  5,  6.  How  is  the 
rate  known  when  it  is  not  mentioned?  7.  How  many  dollars  are  due  for  the 
use  of  $1,000  for  1  year,  when  the  rate  is  6  per  ceitt.?  7.  What  is  the  proper 
name  for  each  of  these  terms  ?  9. 

Case  I.     Q.  What  is  the  nile  for  finding  the  interest  for  1  year?  1. 

1  Interest,  Concern;  advantage;  good;  share;  portion.  Regard  to  private 
property  ;  any  surplus  advantage. 

2  Premium.  Award;  compensation;  prize;  bounty;  sometimes  synonymous  with 
interest. 

3  Definite.    Limited  ;  determinate ;  certain  ;  fixed  ;  exact. 

4  Period.  Circuit ;  epocli ;  a  definite  portion  of  time  ;  end;  stop;  time  between 
one  occurrence  and  another. 

5  Per  Annum,  from  the  Latin, jpcr^  by,  and  annum,  a  year;  hence  the  origin  of 
annually,  meaning  yearly. 

6  Usury.  Formcrli/,  interest ;  note,  illegal  interest;  a  premium  greater  than  that 
allowed  by  law. 

7  Legal.     Lawful ;  according  to  law. 

*  In  some  of  the  8outh('rn  Staics,  it  is  not  considered  usury  to  receive  a  much  high- 
er rate  ;  thus,  in  Louisiana  10  per  cent,  is  the  usual  rate. 


172  ARITHMETIC. 

figures  in  the  product,  for  cents  or  decimals^  than  there  are  cents  or 
decimals  in  the  given  sum. 

2.  What  is  the  interest  of  S5000  at  C  per  cent,  for  one  year  "^ 
What  is  the  amount  1 

$  5  0  0  0  Principal.  $  5  0  0  0  Principal. 

6  Rate.  3  0  0  Interest. 

^.$300,00  Interest.  A.  $  5300  Amount. 


3.  What  is  the  interest  of  $8000  for  1  year  at  6  per  cent. "?  What 
is  the  amount  ?  A.  Interest  $480 ;  Amount  $8,480. 

4.  What  is  the  interest  of  $1200  for  1  year  at  5  per  cent.  1 — at  4 
per  cent.  '?— at  3  per  cent.  1  A.  $60 ;  $48 ;  $36. 

5.  What  is  the  amount  of  $500  for  1  year  at  7  per  cent.? — at  8  per 
cent.  1— at  5|  per  cent.  V  A.  35;  $40  ;  $27.50. 

6.  What  is  the  interest  of  $650.62,  and  of  $453,625,  for  1  year  at 
7  per  cent.  ? 

$650.62  $453,025       A.  $45. 54c.  3Am. 

7  7        A.  $31. 75c.  3-^\m. 

$  4  5.5434       $  31,75375    See  rule  for  pointing  off. 

7.  What  is  the  interest  of  $3019.20  for  1  year  at  6  per  cent.? — at 
5  per  cent.?  A.  181.152;  $150.96. 

8.  What  is  the  amount  of  $1250.375  for  1  year  at  3^  per  cent.? — 
at  5f  per  cent.?  A.  $1294.138+;  $f322.271+. 

CASE   II. 

To  find  the  interest  for  more  years  than  one. 

RULE. 

1.  Multiply  the  interest  of  one  year,  found  as  before,  by  the  number 
of  years. 

2.  What  is  the  interest  of  $375,875  for  3  years  at  6  per  cent.? 
The  interest  for  3  years  must  be  3  times  as  much  as  for  1  year ; 
thus,  $375,875  x  6  per  cent,  x  3  years=$67.6575,  Ans. 

What  is  the  interest  and  amount — 

3.  Of  $200  for  2  years  at  5  per  cent.?  A.  $20  ;  $220. 

4.  Of  $1,700  for  5  years  at  8  per  cent.?         A.  $680  ;  $2,380. 
6.  Of  $7.25  for  2  J-  years  at  4  per  cent.?        A.  $.725  ;  $7.97|. 

6.  Of  44  cents  for  15  years  at  10  per  cent.?      A.  $.66;  $1.10. 

7.  Of  62ct.  5  mills  for  12  years  at  5  per  cent.?     A.  $.37|;  $1. 

8.  Of  je400  for  3  years  at  5  per  cent.?  A.  £60 ;  je460. 

CASE   III. 

To  find  the  interest  for  years  and  months  when  the  rate  is  6  per  cent. 

Q.  What  is  the  interest  of  $400  for  1  year  at  2  per  cent.? — at  5  per  cent.? — 
7  per  cent.? — 10  per  cent.? — 5^^  [)er  cent.?  At  8  per  cent,  what  will  be  the 
amount  of  $200  for  1  year?— of^^SOO?— of  $500?— of  $600? 

Case  II.  What  is  the  rule  for  finding  the  interest  for  two  or  more  years?  1. 
What  is  the  amount  of  $500  at  6  percent,  for  1  year? — for  2  years? — for  3 
years  ? — for  4  years  ? — for  5  years  ? 


LVTERESt.  It3 

RULE. 

1.  Multiply  hy  half  the  number  of  months,  and  cut  off  two  figures) 
as  befote. 

2.  For  at  6  per  cent,  for  every  12  months,  the  rate  for  any  period 
of  time  is  just  half  the  months  of  that  time. 

3.  What  is  the  interest  of  $400  at  6  per  cent,  for  4mo.1— for  lY. 
6mo.1— for  2Y.  Smo."? 

For   4  mo. -^2=2, 

$400  $400  $400        rate;  lY.  6  mo.  =  18 

. 2  9  I  3|     mo.H-2=9,rate;  2Y. 

A.  $8. 00  Ji.  $36. 00  ^.$54. 00        3  mo.=27  mo.-^2  = 

J  2^^  ^^^^ 

4.  Recollect,  when  no  mention  is  made  of  the  rate  of  interest,  6 
per  cent,  is  understood. 

What  is  the  interest  and  amount — 
6.  Of  $600  for  10  months]  A.  $80;  $630 

6.  Of  SI, 600  for  8  months?         .  ^.$64;  $1,664 

7.  Of  $34,675  for  6  months'!  A.  $1.04;  $35,715] 

8.  Of  $13,000  for  lY.  3  months'?  A.  $975;  $13,975 

9.  Of  .$375.50  for  2 Y.  3  months!  A.  $50,691;  $42G.19| 

10.  Of  $689.30  for  12Y.  3mo.?    A.  $506,635+  ;  $1195.935-f 

11.  Of  $313.06  for  lY.  11  months]  A.  $36+  ;  $349.06. 

12.  Of  £500  for  2Y.  1  month]  A.  £62.  10s.;  £562.  10s 

13.  Of  17,000  eagles  for  3Y.  5  months  ]  A.  3,485E.;  20,485E 

14.  Of$595.38  for  1  month]  A.  $2,976+;  $598,356  +  . 

CASE    iV. 
To  find  the  interest  for  days  when  the  rate  is  6  per  cent. 

1.  Since  any  number  of  days  is  a  certain  part  of  a  month,  half  thia 
part  will  of  course  express  the  rate  per  cent,  for  the  days. 

RULE. 

2.  First  reduce  the  days  to  the  fractional  part  of  a  month  (=  30 
days) ;  next  half  this  fraction,  for  the  rate  per  cent,  for  the  days, 
with  which  multiply,  and  point  off  as  before. 

3.  Recollect  that  a  fraction  is  halved  either  by  dividing  its  nume- 
rator, or  by  multiplying  its  denominator,  by  2. 

4.  What  is  the  interest  of  $60  for  15  days]  $  6  0 
15  days=^mo.-^2=|,  the  rate  or  multiplier.  | 

A.  $.15 


5.  What  is  the  amount  of  $200  for  15  days  t  A.  $200.50. 

6.  What  is  the  amount  of  $800.40  for  15  days]     A.  $802.401. 

Case  III.  When  the  time  consists  of  years  and  months,  and  the  rate  is  6  per 
cent.,  how  do  you  proceed?  1.  What  is  the  interest  of  $200  for  1  year  and  8 
months? — for  2  years  6  months  ? — for  3  years  4  months  ?  When  no  per  cent,  is 
mentioned  in  this  work,  what  per  cent,  is  understood  ?  4. 

Case  IV.     Q.   What  is  the  rule  for  days  when  the  rate  is  6  per  cent.?  1. 
Why?  2.     How  is  a  fraction  halved?  3,     What  is  the  interest  of  $60  for  15 
days? — ^for  10  days? — for  5  days? — for  6  days ? — for  1  day? 
15* 


174 


ARltHMfiTlC. 


7.  What  is  the  interest  of  $240  for  10  days?  $240 

10  days =^mo.  -^2 =1,  the  rate  and  multiplier.  |- 


A.  $  .  4  0 


8.  What  is  the  amount  of  $3,000  for  10  dayst  A.  $3005. 

9.  What  is  the  amount  of S480.60  for  lOdays  1  A.  $481,401. 

10.  What  is  the  interest  of  $120.00  for  20  days  1  $12  0.60 
20  days=f^mo.=fmo 


J 

A.  $.4  02  0 


11.  What  is  the  amount  of  $40. 80  for  20  days?       A.  $40,936. 

12.  What  is  the  amount  of  $3678. 90  for  20  days T  A.  $.3691.163. 

13.  What  is  the  interest  of  $360.60  for  19  daysl     $360.60 
19  days==  fn-mo.  -^2=  |f ,  the  rate.  || 

A.  $1.14  19 

14.  What  is  the  amount  of  $420.50  for  19  days  ?  A.  $421.63+. 

15.  When  the  days  are  an  e?jen  number,  it  will  be  more  convenient 
to  find  what  part  of  a  month  half  their  number  is,  for  the  rate  ;  thus^ 
for  20  days  say  10  days  are  ^,  which  is  the  rate  for  20  days. 

16:  What  is  the  interest  of  $24.42|  for  6  days  ?  Half  the  days  is  3 ; 
8d.=  ^  or  yVmo.;  then  y\  is  the  rate.  A.  $.024425=  2ct.  4m.  + 

17.  What  is  the  interest  of  $2.442y%  for  0  days  ?     A.  2yVm.-f 

18.  What  is  the  interest  of  $.24425  for  6  days'?  A.  $.000/^+. 

19.  What  is  the  amount  of  $600  for  20  days  ?  A.  $602. 

20.  WTiat  is  the  interest  of  $60  for  29  daysl — for  28  daysl — for 
27  days  "I— for  26  days  1 — for  25  days  1 — for  24  days  ? — for  23  days  1— 
for  22  days  ?— for  21  days  ? 

A.  $.29;  $.28;  $.27;  $.26;  $.25;  $.24;  $.23;  $.22;  $.21. 

21.  What  is  the  interest  of  $1.20  for  20d.?— for  19d.?— for  18d.'?— 
for  17d.?— for  led.?— for  15d.1— for  14d.?— for  13d.'?— for  lid.'? 

A.  4  miUs;  3y«^m.;  Sy^m.;  d^^m.;  3i%m.;  3m.;  2y«7m.;  2/<^m.; 
2t>. 

22.  What  IS  the  interest  of  $960  for  lOd.1— for  9d.'?— for  8d.?— 
for  7d.'?— for  6d.?— for  5d.'?— for  4d.?— for  3d.'?— for  2d.?— for  Id.'? 

A.  $1.60;  $1.44;  $1.28;  $1.12;  $.96;  $.80;  $.64;  $.48; 
$.32;  $.16. 

CASE  V. 

To  compute  the  interest  for  years,  months,  and  days. 

RULE. 

1.  Calculate  the  interest  for  the  days  as  above,  and  the  interest /of 
the  years  and  months  by  case  iii.,  then  add  both  together. 

Q.  When  the  days  are  an  even  number,  what  different  method  is  suggested  ? 
15.  What  is  the  interest  of  $120  for  10  days?  Of  $1.20  for  10  days?  Of 
$2.40  for  15  days ?— for  20  days ?— for  25  days? 

Case  V.  Q.  To  find  the  interest  of  $600  for  1  year  6  months  15  days,  how 
do  you  obtain  the  multiplier  ?  2.    What  is  the  rule  for  such  cases  ?  1. 


INTEREST.  175 

$  6  0  0 


2.  What  is  the  interest  of  $600  for  1  year, 

6  months,  and  15  days^  15  0 

Say  lY.  6mo.=  18mo-^-2=9;  and  15d.= 
Jmo.->2=5- ;  the  whole  rate  then  is  9j.  • 


5  4  0  0 
A.  S  5  5.5  0 

What  is  the  amount — 

3.  Of  $1,800  for  2  years  6  months  and  10  days  ?        A.  $2,073. 

4.  Of  $15,000  for  1  year  4  months  and  25  days?  A.  $16,262-50. 
6.  Of  $6,000  for  2  years  3  months  and  12  days  t        A.  $6,822. 

6.  Of  $9,030  for  3  years  8  montb-^  and  2  days  ?  A.  $11,019.61. 

7.  Of  $1,260  for  5  years  and  25  days  1  A.  $1,643.25. 

8.  Of  $1,320  for  3  months  and  11  days'?  A.  $1,342.22. 

9.  Of  $60  for  2  months  and  1  day  1  A.  $60.61. 

10.  Of  $30  for  1  month  and  1  dayl  A.  $30,155. 

11.  Of  $3  for  7  years  7  months  and  7  days?  A.  $4,368-}^. 

12.  Of  60  cents  for  10  years  10  months  and  10  days  ?  A.  $.991. 

13.  Of  30  cents  for  3  years  3  months  and  3  days?    A.  $.358+. 

14.  Of  $1,625  for  1  day?  A.  $1,625/^+. 

15.  Of  $300  for  5  years  5  months  and  5  days  ?         A.  $397.75. 

16.  What  is  the  interest  of  $600  for  I  year  7  months  and  15  days  ? 
lY.  7m.  15d.  =  l9imo.-4-2=9f  percent-  A.  $58.50. 

17.  Hence,  we  may  take  half  the  months  and  days  together,  or 
separately,  as  is  most  convenient- 

18.  What  is  the  interest  of  $2,400  for  3  years  9  months  and  10 
days?    45lmo.-^2=22f,  the  multiplier.  A.  $544. 

19.  What  is  the  amount  of  $40.80  for  5  years  3  months  and  7  days  ? 
633^-^2=3111  per  cent.  ^.53.699^. 

20.  What  is  the  amount  o/$300  for  2  years  8  months  and  5  days? 
32imo.-^2=163^  per  cent  A.  $348.25. 

21.  What  is  the  amount  of  $1,200  for  12  years  11  months  and  29 
days?  A.  $2,135.80. 

Note. — Few  examjies  occur  more  difficult  than  the  last,  and  it  is 
solved  with  one-thirt?  the  usual  number  of  figures  required  by  other 
methods. 

CASE  VI. 

To  compute  the  interest  between  different  dates. 

RULE. 

1.  Write  down  the  later  year>,  and  on  the  rights  the  months  and 
days  of  that  year  that  have  elapsed ;  do  the  same  with  the  other  date. 

2.  Subtract  the  earlier  from  the  later  date,  as  in  Compound  Sub- 
traction^ always  reckoning  30  days  to  the  month,  and  12  months  to 

Q,  What  is  the  multiplier  for  2  years  6  months  10  days?— for  lY.  4m.  25d.? — 
for  2Y.  3m.  12d.?— for  2Y.  6mo.  5d.?  What  is  the  interest  of  $30  for  1 Y.  4mo. 
15d.?— for  lY.  6mo.  lOd.?— for  2Y.  6mo.  and  5d.? 

Case  VI.  Q.  Describe  the  whole  process  m  casting  the  interest  between 
different  dates.  1.  3. 


176  ARITHMETIC. 

the  year ;  the  remainder  will  he  the  required  time,  with  which  proceed 
as  before. 

3.  What  is  the  interest  of  $180  from  September  20th,  1836,  to 
April  5th,  1839  \ 

In  1839,  3  months  [viz.  Jan.  Feb. 

March]  and  5  days  have  elapsed  ;  and 

Years.     '  mo.       d.       there  have  elapsed  in  1836,  8  months 

*  1  Q  ?  n  ■  o  '  nn  and  20  days.  Subtract,  saying  20d. 
^  ^  -^  ^  ^  ^Q  from  30d.  leaves  lOd.,  and  5d.-15d. 
Time   2  .    6    .   15        &c.  Next  find  the  interest  of  the  $180 

for  the  2Y.  6mo.  and  15d.  as  before. 
A.  $27.45- 

4.  Wliat  is  the  interest  of  $320.40  from  March  27,  1827,  to  Feb- 
ruary 12, 1828? 

5.  Note. — Accoiding  to  the  table  at  the  bottom  of  the  page,  if  the 
date  were  February  20th,  for  instance,  it  would  be  expressed  1828, 
Imo.  20d.;  but  as  it  is  the  12th  instead  of  20th,  call  the  later  date 
1828,  Im.  12d.  Against  March  in  the  same  table  we  find  2m.  20d.; 
then  call  the  earlier  date  1S27,  2m.  27d.  A.  $16,821. 

6.  What  is  the  interest  tf  $1,640  from  July  5th,  1826,  to  June 
20th,  1828?  A.  $192.70. 

7.  What  is  the  interest  of  ^15.30  from  April  1st,  1828,  to  Sep- 
tember 11th,  1830?  ^.$00.9106. 

8.  What  is  the  interest  of  $840  from  June  11th,  1820,  to  April 
20th,  1822  ?  A.  $93.66. 

9.  What  is  the  interest  of  $60.50  from  March  1st,  1819,  to  No- 
vember 21st,  1825?  ^.$24.4016.+ 

10.  What  is  the  interest  of  $240  front  October  31st,  1832,  to  July 
5th,  1834? 

Y.  m.         d.        Since  30d.  in  reckoning  interest =lm-, 

18  3  4.6.  5  then  31d.  =  lm.  Id.;  therefore  add  Im.  to 
18  3  2-  9  .  31  9ni.  =  10m.,  and  suUract  10m.  Id.  instead 
Time  1    .8.4     of  9m.  3 Id.  ^.$24.16. 

IJ.  What  is  the  interest  of  $840.60  from  May  31st,  1828,  to  June 
25,1830?  4.  $104.234tV 

12.  What  is  the  interest  of  $120.50  from  December  1st,  1815,  to 
May  31,  1820  ?  The  required  time  is  4Y.  5m.  30d.,  but  as  30d.  =1 
month,  call  the  whole  4Y.  6  months.  A.  $32,535. 

13.  What  is  the  interest  of  $75.80  from  August  31,  1827,  to  July 
Ist,  1830? A.  Time,  2Y.  lOmo.;  Int.  $12.88-^^. 

Q.  How  is  April  5th,  1839,  and  September  20th,  1836,  written  downf  3. 
In  taking  1832Y.  9m.  31d.  from  1834Y.  6m.  5d.  how  is  the  31d.  subtracted?  10. 

*  Suppose  the  date  of  each  month  in  the  year  were  the  20th,  for  instance,  the  time  that 
has  elapsed  would  be  expressed  as  follows,  viz. 

January   Om.  20d.  May      4m.  20d.  September  8m.  20d. 

February  im.  20d.  June      6m.  20d.  October        9m.  20d. 

March      2m.  20d.  July      6m.  20d.  November  10m.  20d. 

\  April        8m.  20d.  August  7m.  20d.  December  11m.  20d 


INTEREST. 


177 


14.  What  is  the  interest  of  $75.80  from  March  1st,  1836,  to  May 
31,  1840?  -4.  $19,329. 

15.  What  is  the  amount  of  $14.30  from  December  25th,  1837,  to 
January  19th,  1840?  A.  $16.073/g-. 

16.  What  is  the  amount  of  $715.20  from  November  10th,  1827,  to 
February  20th,  1828?  A.  $727.12. 

17.  What  is  the  amount  of  $50.60  from  February  14th,  1829,  to 
August  4th,  1831.  A.  $58,105^-. 

18.  What  is  the  amount  of  $500  from  March  31st,  1830,  to  July 
1st,  1835?  A.  $657.50. 

19.  Suppose  that  A,  on  the  11th  of  October,  1835,  borrows  of  B 
$1700.50,  for  which  he  is  to  pay  interest;  what  does  A  owe  B  on 
the  11th  of  January,  1837?  A.  $l,828.03f. 

20.  Suppose  a  note  of  $300  which  is  on  interest  from  July  5th, 
1836,  is  paid  June  17th,  1838  ;  what  sum  of  money  will  cancel  the 
debt?  A.  $335.10. 

21.  If  a  merchant  borrows  $10,000,  April  1st,  1828,  and  October 
16th,  1830,  pays  $11,000,  how  much  will  he  then  owe,  including  the 
interest?  A.  $525. 

22.  A  and  B,  on  settling  their  accounts,  found  a  balance  due  A 
of  $320,  for  which  B  gave  his  note,  dated  Boston,  July  5th,  1838, 
with  interest.  On  the  11th  of  December,  1840,  B  paid  A  $400; 
how  did  the  balance  stand  then  ?  A.  A  owes  B  $33.28. 

CASE    VII. 

To  compute  the  interest  more  accurately  for  days. 

1.  By  reckoning  30  days  to  the  month  and  12  months  to  the  year, 
we  get  for  360  days  the  interest  for  365,  being  a  gain  of  5  days  inter- 
est in  a  year.* 

2.  Since  5  days  are  ^\  of  lY.,  or  13  days,  for  instance,  -^^  of  lY., 
therefore — 

3.  Multiplying  the  interest  for  1  year,  found  as  before,  by  the  days, 
and  dividing  by  365,  [but  in  leap  year  by  366,]  or  deducting  ^^^  from 
the  interest  computed  by  case  v.,  gives  the  true  interest. 

4.  What  is  the  interest  of  $14,600  for  90  days  at  7  per  cent.? 
14,600 X. 07x90-4-365.  A.  $252. 

5.  What  is  the  interest  of  $438,000  for  300  days  ?  What  the  in- 
terest for  [300d.-^30d.=]  10  months,  computed  by  case  v. 

A.  $21,600;  $21,900. 

6.  When  our  national  debt  fell  a  little  short  of  $123,735,000, 

Case  VII.  Q.  W^hat  error  is  there  in  the  last  rule  ?  1,  What  is  the  general 
direction  ?  3.  Why  deduct  i^U,  or  why  divide  by  73  ?  2.  What  are  the  multi- 
plying and  dividing  terms  in  casting  the  interest  on  $4,000  accurately  for  125 
days  ? — for  150  days  in  the  year  1840  ? 

*  The  error  in  small  sums,  however,  is  a  mere  trifle ;  but  in  large  sugis  it  becomes  too 
important  to  be  neglected. 


178 


ARITHMETIC. 


what  would  be  the  difference  between  computingjts  interest  by  the 
present  method,  and  computing  it  by  case  v.,  reckoning  the  rate  at  6 
per  cent,  and  the  time  200  days  ?  A.  $56,500. 

7.  What  is  the  interest  of  $274,500  at  4  per  cent,  for  150  days  of 
a  leap  year  1  A.  $4,500. 

8.  Suppose  the  bank  of  England,  through  which  the  national  re- 
venue, being  about  $210,000,000,  is  annually  collected,  should  receive 
the  interest  on  that  sum  for  292  days,  how  much  would  be  gained  by 
computing  the  interest  by  months  instead  of  by  days  1  A.  $140,000. 

CASE    VIII. 

To  find  the  interest  at  any  rate  per  cent. 

RULE. 

1.  Find  the  interest  for  the  given  time,  as  if  it  were  6  per  cent., 
then  multiply  it  by  the  given  rate  and  always  divide  by  6. 

2.  For  1  per  cent,  is  ^  of  6  per  cent.,  2  per  cent.  |  or  ^,  3  per  cent, 
f  or  I,  &c.* 

3.  What  is  the  interest  for  $600  for  1  year  2  months  and  15  days 
at  5  per  cent.  1 

$  6  0  0  r«       •         r  11 

Or,  smce  5  per  cent,  is  ^  less 

than  6  per  cent.,  we  may  simply 
deduct  I  of  the  interest  of  6  per 
cent,  from  itself,  as  follows, — 

Thus,  6)$43.50  int.  at  6  per  cent. 

7.25 

A.  $36.25int.  at  5  percent. 

4.  What  is  the  interest  of  $240  for  2  years  6  months  at  3  per 
cent?  A.  $18. 

Case  VIII.  Q.  What  is  the  rule  for  finding  the  interest  for  any  period  of 
time  at  any  rate  per  cent.  ?  1.  Why  ?  2.  In  computing  the  interest  of  $600  foi 
lY.  2m.  15d.  at  5  per  cent.,  what  Would  you  multiply  and  divide  by  ?  3.  Con- 
sult the  reference  from  2  at  the  bottom  and  tell  how  the  process  may  be  abbre 
riuted  when  the  rate  is  either  2,  3,  4,  5,  7,  8,  9,  or  12  per  cent.  When  the  prin- 
ciple is  60  dollars  and  the  time  10  days,  what  is  the  interest  at  2  per  cent.  ? — 
at  3  per  cent.  ? — at  4:  per  cent.  ? — at  8  per  cent.  1 — at  12  per  cent.  ? 


7  1 

1 
4  2 

5  0 
0  0 

$  4  3.5  0  at  6  per  cent. 
5  given  rate. 

6 

)  2  1  7 

5  0 

A 

.$36 

.2  5  at  5  per  cent 

*  Hence  when  the  given 
rate  is  not  6  per  cent,  we  may 
find  the  interest  at  6  per  cent. 
as  before,  then  proceed  with 
that  interest  according  to  the 
directions  in  the  adjacent  Ta- 
ble. 


by  6. 
by  3. 

by  2. 


For    1  per  cent.   —  ^  =    —divide 

For    2  per  cent.   =  f  =^  —divide 

For   3  per  cent.   =  |   =|  —divide 

For  4  per  cent.  —  f  ~  f  — deduct 

For  6  per  cent.   =  -f  =      —deduct 

For   7  per  cent,  =  ^  =  1^— add 

For   8  per  cent.   ~  -^  =  1^— add 

For  9  per  cent.    =  f  =  1 2— add 

For  10  per  cent.  =  V  =  ^  §— add  ^ 

For  11  per  cent.  =  -g-  =1^— add  b 

For  12  per  cent.  =  Y=2  —multiply  by  2 


INTEREST.  179 

5.  Of  $360  for  1 Y.  8m.  at  5  per  cent.  1  A.  $30. 

6.  Of  $480  for  5Y.  4m.  at  1  per  cent.  ^  A.  $25.60. 

7.  Of  $6.60  for  lY.  8m.  at  7  per  cent.  1  A.  $0.77. 

8.  Of  $3.20  for  3Y.  4m.  at  11  per  cent?  A.  $1,173.+ 

9.  Of  $120  for  8Y.  4.m.  at  12  per  cent.  ?  A.  $120. 

10.  Of  $620  for  2Y.  6m.  at  9f  per  cent.  1  A.  $151.12^. 

11.  Of  $1.20  for  9Y.  2m.  at  11|  per  cent.  ?  A.  $1.2783^7- 

12.  Of  $720  for  lY.  4m.  at  13|  per  cent.  ?  A.  $131.20. 

13.  What  is  the  interest  of  $1,200.60  from  January  1st  to  the 
18th  of  April  following,  at  1 1^  per  cent.  1  A.  $41.037yV  + 

14.  What  is  the  amount  of  $500  from  July  13th,  1833,  to  October 
27th,  1835,  at  5f  per  cent.  ?  A.  $565.804rV  + 

CASE   IX. 

A  concise  method  of  finding  the  interest  at  7  per  cent.,  being  the 
legal  rate  in  the  State  of  New  York. 

RULE. 

1.  Com.pute  the  interest  as  before  for  6  per  cent.,  then  add  ^  of 
this  interest  to  itself.     [See  case  viii.  2.  reference.] 

$  3  6    0 

2.  What  is  the  interest  of  $360  9  J 
for  1  year  6  months  and  15  days  at 
7  per  cent.  1 

»  ) 
The  time  is  18^  months^2=9|^, 
the  rate.  ^   ^ 

3.  What  is  the  interest  of  $60  for  2  years  4  months  at  7  per 
cent.  1  A.  $9.80. 

4.  What  is  the  amount  of  $120.60  for  1  year  6  months  10  days  at 
7  per  cent.  1  A.  S133.497t\. 

5.  What  is  the  amount  of  $241.20  at  7  per  cent,  for  6  months  20 
days  1— for  Im.  Id.  1— for  1 Y.  4m.  5d.  T— for  2Y.  6m.  25d.  ? 

A.  $250.58;  $242,653;  $263.946 ;  $284,582. 

6.  What  is  the  difference  between  the  legal  interest  of  $,1(0,090  in 
New  York,  and  the  legal  interest  of  the  same  sum  in  New  England, 
from  April  1st,  1836,  to  October  1st,  18401  A.  $450. 

CASE    X. 

To  find  the  Principal  at  interest. 

1.  What  is  the  interest  of  $1  for  1  year  8  months  ?  A.  10  cents. 

2.  If  the  interest  of  $1  for  lY.  8m.  is  10  cents,  what  must  be  the 
interest  of  $2  for  the  same  time  1  What  of  $3 1— of  $5  1— of  $75 1 
—of  $819.75?  A.  $.20;  $.30;  $.50;  $7.50;  $81.97|. 

3.  If  the  interest  of  $1  is  10  cents,  what  sum  will  be  required  to 

Case  IX.  Q.  What  is  the  rule  for  the  State  of  New  York  in  which  the  rate 
is  7  per  cent.  ?    1. 


3  2 

9  0 
4  0 

)  3  3. 
5. 

3  0 
5  5 

13  8, 

.8  5 

180  ARITHMETIC. 

draw  20  cents  interest  1    What  to  draw  30  cents'? — 50  cents'? — 
11^7.50 1--$81.97i?  A.  $2;  ^3;  $5;  $75;  $819.75. 

RULE. 
4.  Divide  the  given  interest  by  the  interest  of  $1,  for  the  given  rate 
and  time,  the  quotient  will  he  the  required  principal. 

6.  What  sum  of  money  put  at  interest  1  year  and  8  months  will 
gain  $20.60? 

Note.— Divide  $20.60  by  the  interest  of  $1  for  lY.  8m.  A.  $206. 

7.  What  principal  will  gain  $200  in  4Y.  2m'?  A.  $800. 

8.  Suppose  a  gentleman's  income  from  his  property  is  $4,800  per 
annum,  how  much  is  he  worth  1  A.  $80,000. 

9.  Suppose  the  interest  of  a  certain  sum  for  2Y.  6m.  is  $300  ; 
what  is  that  sum  ■?  A.  $2,000. 

10.  What  is  that  sum  the  interest  of  which  at  7  per  cent,  for  5Y. 
4m.  is  $728.  A.  $1,950. 

11.  What  sum  is  that  whose  interest  is  $200,104  yV  for  10  years 
9  months,  at  4  per  cent. '?  j4.  $465.36. 

12.  Suppose  a  note  which  was  dated  February  15th,  1828,  had 
accumulated  $60  interest  on  the  15th  of  August,  1830 ;  what  was  the 
principal  of  the  note "?  A.  $400. 

13.  A  merchant  on  a  note  receivable  dated  May  5th,  1820,  cal- 
culated the  interest  up  to  September  20th,  1823,  the  time  of  its 
maturity,  and  made  it  $3,645,  noting  it  down.  When  the  debtor 
called  to  pay  the  note,  it  could  not  be  found ;  can  you  tell  what  was 
the  face  of  the  note,  and  what  sum  ought  to  be  considered  full  pay- 
ment of  the  same. 

A.  $18,000  principaH-$3,645=$21,645  full  payment. 

CASE    XI. 

To  find  the  rate  per  cent. 

RULE. 

1.  Divide  the  given  interest  hy  the  interest  of  the  principal  at  1 
per  cent,  for  the  given  time,  the  quotient  will  be  the  rate  per  cent. 

2.  For  the  required  rate  is  of  course  as  many  times  greater  than 
1  per  cent,  as  the  given  interest  is  greater  than  the  interest  at  1 
per  cent. 

3.  What  is  the  interest  of  $2,000  for  1  year  at  1  per  cent.  ■? — at 
6  percent.'?  A.  $20;  $120. 

4.  If  $20  interest  is  gained  in  one  year  on  $2,000  ;  what  per  cent, 
will  gain  six  times  as  much — ^that  is  $120  in  the  same  time  1 

$120-^20=6  per  cent.  A. 

Case  X,  What  is  the  rule  for  finding  the  principal  ?  4.  When  the  interest 
is  $20.60  and  the  time  lY.  8m.,  how  is  the  principal  found?  5.  What  is  a  gen- 
tleman worth  when  his  interest  money  is  $2,400  per  annum?  When  it  is 
$3,600  per  annum  ? 

Case  XI.  What  is  the  rule  for  finding  the  rate  per  cent.?  1.  What  is  the 
reason  of  the  rule  ?  2.  When  $60  dollars  are  paid  as  the  interest  of  $600  for 
1  year  and  8  months,  what  must  be  the  rate  of  aterest  ? 


INTEREST.  181 

5.  What  per  cent,  is  $3  645  interest  on  $18  000  for  3  years  4 
months  15  days?  '         A.  6  per  cent. 

6.  Suppose  a  note  of  $2  918  has  acquired  from  February  1st,  1825, 
to  December  16th,  1827,  $587.24^,  what  must  have  been  the  rate 
per  cent.  ?  A.  7  per  cent. 

CASE    XII. 

To  find  the  time  in  interest. 

1.  What  is  the  interest  of  $200  for  one  year  ?  A.  $13. 

2.  If  the  interest  of  $200  for  one  year  is  $12,  howmany  years  will 
be  required  to  gain  $24 1  A.  24-^12=2  years. 

3.  Therefore,  as  many  times  as  the  interest  of  1  year  is  contained 
hi  the  given  interest,  so  many  years  will  be  required  to  gain  the  true 
interest. 

RULE. 

4.  Divide  the  given  interest  by  the  interest  of  the  principal  for  one 
year  at  the  given  rate. 

5.  How  long  will  $600  be  in  gaining  $72  T  A.  2  years. 

6.  How  long  will  $1,000  be  in  gaining  $120 1— $180  % — $240  1 — 
$3001— $3601  A.  2Y.  3Y.  4Y.  5Y.  6Y. 

7.  What  time  will  be  required  for  $356.76  to  gain  $53.5141 

A.  2  years  6  months. 

8.  How  long  will  $1,800  be  in  gaining  $702 1  A.  6Y.  6m. 

9.  What  time  will  be  required  for  $2,500  to  gain  $470  at  8  per 
cent.  1  A.  2Y.  4m.  6d. 

10.  Suppose  I  receive  $157  as  the  lawful  interest  of  $1,200  for  a 
certain  time,  what  is  that  period  of  time  1  A.  2Y.  2m.  5d. 

CASE    XIII. 

To  find  the  date  from  which  the  interest  begins  or  ends,  that  is,  to 
find  the  later  or  earlier  date. 

1.  When  will  $5,000  on  interest  from  October  6th,  1830,  gain 
S762.50. 

2.  The  time  found  by  the  last  case  is  2Y.  6m.  15d. 
1830Y         9  6d  Since  3m.  2  Id.  of  the  year  1833 

2  Y         6  m        15  d      ^^^'^  elapsed,  it  brings  the  year, 

,    Q  o  o  V ^~~ S1~T    by  the  reference  in  case  vi.  3.  to 

1  b  3  3   y.        3  m.       21  d.    ^prii2ist.     A.  April 21st,  1833. 

3.  To  find  the  earlier  date,  we  may  subtract  the  time  between  the 
two  dates,  from  the  later  date,  thus, 

April  21st,  1833  =  1833Y.     3m.  21d.=  the  later  date. 

2Y.     6m.  15d. =the  time  between  the  dates. 
A.  1 830  October  6  =  1 830Y\    9m!     ed".  =  the  earlier  date. 

RULE. 

4.  Having  found  the  intermediate  time  by  the  last  'case  add  it  to 

Case  XII.     Q.     What  is  the  rule  for  finding  the  time  ?  4.    Why  ?  3.    What 
time  will  be  required  for  $500  to  draw  $30  interest  ?— to  draw  $45  ?— $601 
16 


18^ 


ARITHMETIC. 


the  earlier  date  for  the  later  date,  and  subtract  it  from  the  later  date 
for  the  earlier  date. 

6.  Paid,  July  5th,  1838,  $48  interest  on  a  note  of  $000 ;  from  what 
date  did  the  interest  commence  1  A.  March  5th,  1837. 

6.  Suppose  a  note  of  $600  is  dated  March  5th,  1837,  when  will 
the  interest  amount  to  $48  ?  A.  July  5th,  1838. 

7.  Suppose  I  pay  $42,  August  23d,  1833,  it  being  the  lawful  in 
terest  on  a  note  of  $250,  what  is  the  date  of  the  note  ] 

A.  June  11th,  1836. 

8.  On  the  5th  of  June,  1820,  a  merchant  paid  $260,  the  interest  op 
a  note  of  $1000 ;  when  did  the  interest  commence  \ 

A.  February  5th,  1816. 

9.  If  a  man  tells  you  he  holds  in  his  hand  a  note  of  $3,600,  the  in 
terest  of  which  from  its  date  to  Oct.  29,  1830,  at  7  per  cent.,  amounts 
to  $1,438-2,  and  requests  you,  without  seeing  the  note,  to  name  its 
date,  could  you  do  it?  A.  February  14th,  1825. 


INTEREST  ON  NOTES. 

LXXXI.  1.  A  Note  is  a  written  promise  to  pay  a  certain  sum 
of  money,  or  its  value  in  goods,  on  demand,  that  is,  when  demanded ; 
or  at  some  future  day,  and  hence,  all  notes  are  called  promissory  notes. 

2.  A  Negotiable  Note  is  one  which  is  made  payable  to  A.  B., 
or  order. 

3.  By  indorsing  a  note  is  understood  that  the  person  to  whom  it  is 
made  payable  writes  his  name  on  the  back  of  it,  and  thereby  becomes 
responsible  for  its  payment. 

4.  When,  however,  a  note  is  not  paid  at  maturity,  the  responsi- 
bility of  the  indorser  ceases  unless  he  be  notified  of  the  fact  within 
the  time  prescribed  by  law. 

5.  A  person  holding  a  negotiable  note  may  transfer  it  by  indorsing 
it,  and  whoever  buys  it  may  lawfully  demand  payment  of  the  signer 
of  the  note,  and  if  the  signer  refuses,  from  inability  or  otherwise,  to 
pay  it,  the  purchaser  may  lawfully  demand  payment  of  the  indorser. 

6.  If  a  note  be  made  payable  to  A.  B.,  or  bearer,  then  the  signer 
only  is  responsible  to  any  one  who  may  purchase  it. 

7.  Unless  a  note  be  written  payable  on  some  specific  future  time, 
it  should  be  written  "  on  demand,"  but  should  the  words  "  on  de- 
mand," be  omitted,  the  note  is  supposed  to  be  recoverable  by  law. 

8.  When  the  words  "  with  interest,"  are  omitted,  a  note  is  not  sup- 
Case  XIII.     How  is  the  dates  from  which  the  interest  begins  or  ends  ob- 
tained ?  4.     How  far  will  the  3  first  months  and  21  days  carry  the  year? 

LXXXI.  Q.  What  is  a  note?  1.  When  is  a  note  negotiable?  2.  What 
does  indorsing  a  note  imply  ?  3.  Is  the  indorser  holden  after  it  is  due  ?  4.  How 
is  a  negotipble  note  transferred  ?  5.  When  is  the  signer  only  responsible  ?  6. 
When  shoujv,  „  liote  be  written  "  on  demand?"  7. 


INTEREST.  183 

posed  to  be  on  interest.  Except  when  a  note  payable  at  a  future 
day  becomes  due  ;  it  is  then  considered  on  interest  from  that  time  till 
paid,  though  no  mention  be  made  of  interest. 

9.  No  mention  need  be  made  of  the  rate  of  interest ;  that  particular 
is  settled  by  law,  and  will  be  collected  according  to  the  laws  of  the 
State  where  the  note  is  dated. 

10.  If  two  persons  jointly  and  severally  sign  a  note,  it  may  be  col- 
lected by  law  of  either.  A  note  is  not  valid,  unless  the  words  "  for 
value  received,"  be  expressed. 

11.  When  a  note  is  given  payable  in  any  article  of  merchandise  or 
property  other  -than  money,  deliverable  on  a  specified  time,  such 
articles  should  be  tendered  in  payment  at  said  time  ;  otherwise,  the 
holder  of  the  note  may  lawfully  demand  the  value  in  money. 

12.  The  person  who  gives  a  note  is  called  the  Signer  or  Drawer, 
and  when  the  note  is  indorsed  by  a  third  person,  the  Principal,  be- 
cause the  holder  of  the  note  looks  ^r*^  to  him  for  payment. 

13.  The  sum  or  debt  for  which  a  note  is  given,  is  called  the  prin- 
cipal or  FACE  of  the  note ;  the  person  indorsing  it  the  Indorser,  and 
the  person  to  whom  it  is  indorsed  when  sold,  the  Indorsee  or  As- 
signee. 

14.  When  a  partial  payment  of  a  note  is  made,  the  creditor  specifies 
in  writing  on  the  back  of  the  note,  the  sum  paid,  and  the  time  when 
it  is  paid,  acknowledging  it  by  subscribing  his  name,  which  is  then 
called  an  Indorsement.    Interest  when  paid,  is  indorsed  as  such  in 

like  manner. 

CASE   I. 

When  there  are  no  indorsements. 

RULE. 

15.  Find  the  interest  of  the  note  from  its  date^  or  the  time  when 
lie  interest  commenced  up  to  the  tiine  it  is  due  or  paid ;  then  add- 
ing this  interest  to  the  face  of  the  note,  will  give  the  sum  due,  {as 
in  case  v.) 

(16.)    $500.  Boston,  January  1,  1836. 

For  value  received,  I  promise  to  pay  William  Marshall 
five  hundred  dollars  in  six  months,  with  interest. 

A.  $515  due  July  1,  1836.  Peter  Stiver. 

(17.)     $800.  New  York,  April  1st,  1828. 

For  value  received,  I  promise  to  pay  Peter  Parley,  Esq. 
eight  hundred  dollars,  in  two  years  with  interest.     [See  lxxx.  6.] 
A.  $912  due  April  1,  1830.  S.  G.  Goodrich. 

Q.  Are  all  notes  on  interest  ?  8.  Need  the  rate  of  interest  be  specified  ?  9. 
What  is  said  of  a  note  given  by  two  persons  jointly  and  severally  ?  10.  What 
of  the  words  "for  value  received?"  10.  AVhat  of  articles  specified  to  be  de- 
livered? 11.  What  do  you  understand  by  the  Signer  or  Drawer  of  a  note?  12. 
By  the  Principal,  and  why?  12.  By  the  Indorser?  13.  By  the  Indorsee  or 
Assignee  ?  13.  By  the  face  or  principal  of  the  note  1  13.  How  is  an  indorse- 
ment made  ?  14. 


184 


ARITHMETIC. 


(18-)  Providence,  June  10,  1826. 

For  value  received,  I  promise  to  deliver  unto  John  Northam 
two  bales  of  good  cotton,  each  bale  to  contain  four  hundred  pounds, 
[valued  at  12  cents  per  pound]  on  or  before  the  twenty-fifth  day  of 
October,  eighteen  hundred  and  twenty-eight.       Stephen  Trader. 

The  above  cotton  was  not  tendered  in  payment  till  October  25th,  1829,  when 
It  had  declined  25  per  cent,  and  was  therefore  refused,  [see  11.]  but  when  the. 
note  came  to  maturity  it  had  advanced  15  per  cent.  ;  now  what  would  the 
debtor  have  lost  if  he  had  furnished  the  cotton  according  to  agreement,  and  hov/ 
much  does  the  creditor  gain  by  refusing  it  when  tendered  ?  [See  8  and  11.]  * 
A.  Debtor's  loss  $14.40  ;  Creditor's  gain  $29.76. 

CASE   II. 

When  there  is  only  one  indorsement  or  payment. 

19.  When  a  settlement  is  made  within  a  short  time  from  the  date 
or  commencement  of  interest,  it  is  generally  the  custom  to  proceed 
according  to  the  rules  in  the  two  following  cases. 

RULE. 

20.  Find  the  amount  of  the  principal  for  the  whple  time,  also  the 
amount  of  the  payment  from  the  time  it  was  paid  to  the  time  of  settle- 
ment ;  then  deduct  the  amount  of  the  payment  from  the  amount  of  the 
principal,  the  remainder  will  be  the  balance  due. 

(21.)    $200.  Boston,  July  1,  1838. 

For  value  received,  I  promise  to  pay  William  Grey,  or 
order,  two  hundred  dollars  on  demand,  with  interest. 

Joshua  Huckins. 
On  this  note  there  was  the  following  indorsement.    Received,  December  16th, 
1838,  seventy-five  dollars.     What  was  the  balance  due  March  16th,  1839  ? 

Principal, $200.00 

Interest  from  July  1st,  1838,  to  March  16th,  1839,        -        -        -  8.50 

Amount  of  principal  for  83- months $208,50 

Payment, $75.00 

Interest  from  Dec,  16,  1838,  to  March  16,  1839,       -        -        1.12| 

The  amount  of  payment  for  3mo.  deducted,  -        -        -        ~$  76.12^ 

Balance  due  March  16,  1839, Answer,  $132.37^. 

(22.)    $1,800.  Boston,  September  10th,  1830. 

For  value  received,  I  promise  to  pay  John  Frothing, 
or  order,  eighteen  hundred  dollars  in  one  year  with  interest. 

Jehiel  Johnson. 

The  following  payment  was  made.  January  25th,  1831,  received  six  hundred 
dollars.    What  was  the  balance  due  July  15th,  1831  ?  A.  $1,274.50. 

Case  II.  Q.  When  there  is  only  one  indorsement  how  do  you  pro- 
ceed? 20. 

*  The  note,  of  course,  does  not  tlraw  interest  till  after  its  maturity,  and  the  debtor's 
loss,  had  he  delivered  the  goods  at  the  time  specified,  would  have  been  15  per  cent,  on 
the  face  of  the  note  =-.$14.40.  The  creditor's  gain  will  be  the  diflerencc  between  the  face 
of  the  note  with  the  interest  accruing  for  one  year  ($101.76,)  and  the  value  of  the  coltoa 
at  the  time  it  was  tendered  ($72.00.) 


INTEREST.  185 

(S3.^    $600.  New  York,  August  8th,  1835. 

For  value  received,  I  promise  to  pay  Messrs.  Brown 
and  Catlin,  six  hundred  dollars,  on  demand,  with  interest. 

Thomas  Tippleton. 
On  this  note  was  the  following  indorsement.     Received,  May  1st,  1836,  four 
hundred  and  fifty  dollars.     What  is  the   sum  due  July  15th,  1836  ?       [See 
LXXX.  6.] A.  $182,841. 

CASE    III. 

When  there  are  two  or  more  indorsements 

RULE. 

24.  Find  the  amount  of  each  payment  and  of  the  principal  as 
before ;  then  deduct  the  total  amount  of  the  payments  from  the  amount 
of  principal. 

(25.)    $300.  Hartford,  April  1,  1825. 

For  value  received,  I  promise  to  pay  Rufus  Stanly,  or 
order,  three  hundred  dollars,  on  demand,  with  interest. 

Simeon  Thompson. 
On  this  note  were  the  following  indorsements.     Oct.  1st,  1825,  received  one 
hundred  dollars  ;  April  16th,  1826,  received  fifty  dollars  ;  Dec.  1st,  1827,  received 
one  hundred  and  twenty  dollars.     What  was  the  balance  due  April  1st,  1828  ? 

Principal, $300.00 

Interest  of  the  principal  to  April  1st,  1828 j      .....    54.00 

Amount  of  principal  for  36  ragnths,  $354.00 

First  payment  Oct.  1st,  1825, $100.00 

Interestto  April  1st,  1828,  (30mo.)   .        .        ^        .        -     15,00 
Second  payment,  April  16,  1826,  ....        50.00 

Interest  to  April  1st,  1828, 5.87 

Third  payment,  Dec.  1st,  1827 120.00 

Interest  to  April  1st,  1828,  (4mo.)     .,       .        .        ^        .       2.40 

Amount  of  payments  deducted,     *        » $293.27 

Balance  due  April  1st,  1828  -        -        -        -        -  Answer,  $60.73 

(26.)    $500.  New  Haven,  July  1st,  1825. 

For  value  received,  I  promise  to  pay  Peter  Trusty,  or 
bearer,  five  hundred  dollars,  on  demand,  with  interest. 

James  Careless. 

Indorsements. — Received,  July  16th,  1826,  two  hundred  dollars.     Received, 
January  1st,  1827,  forty  dollars.     Received,  March  16th,  1827,  two  hundred 
and  thirty  dollars.     What  sum  remained  due  July  16th,  1828? 
Results,  591.25;  516.10.     A.  $75.15. 

(27.)    $1,000.  Portland,  January  16th,  1820. 

For  value  received,  we  jointly  and  severally  promise 
to  pay  to  Stimpson  and  Ripley,  or  order,  one  thousand  dollars,  on 
demand,  with  interest.  William  Bird. 

James  Bement. 

On  the  back  of  this  note  were  the  following  indorsements,  viz. — Received, 
March  16th,  1821,  six  hundred  dollars.     Received,  May  1st,  1822,  one  hundred 
and  twenty  dollars.     Received,  July  16th,  1822,  one  hundred  and  eighty  dollars. 
What  was  the  balance  due  January  16th,  1823? 
Results.  1180;  97650.     A.  $203.50. 

Cass  III.    Q.    When  there  are  several  indorsementa  what  is  the  rule  ?  24. 
16* 


186  ARltHMfitlC. 

(28.)    85,000.  New  YoftK,  June  Ist,  1835. 

Four  months  from  date,  we  jointly  and  severally  pro- 
mise to  pay  Timothy  Dickens  and  Co.  five  thousand  dollars,  with 
interest,  value  received.  James  Rover. 

John  Townsend. 

Indorsements. — Received,  October  1st,  1835,  one  thousand  dollars.  Received, 
January  15th,  1836,  one  thousand  dollars.  Received,  March  26th,  1836,  one 
thousand  dollars.  Received,  April  1st,  1836,  one  thousand  dollars.  "What  was 
the  balance  due  July  I6th,  1836?     [See  Lxxx.  6.] 

Results.  539375;  4l324l3.     Ail261.337. 

29.  The  foregoing  rule  should  be  restricted  to  cases  in  which  set- 
tlement is  made  within  a  year  from  the  commencement  of  interest ; 
for,  beyond  that  period,  its  error,  in  comparison  with  the  legal  rule 
below,  becomes  too  important  to  be  neglected. 

30.  The  rule  below  is  essentially  the  same  as  that  established  by 
the  United  States'  court,  and  adopted  by  most  of  the  federal  courts 
in  the  Union. 

GENERAL  RULE. 

31.  Find  the  amount  of  the  principal  to  the  time  of  the  first  pay- 
ment; subtract  the  payment  from  this  amount,  and  then  find  the 
amount  of  the  remainder  to  the  time  of  the  second  payment ;  deduct 
the  payment  as  before ;  and  so  on  to  the  time  of  settlement. 

32.  But  if  any  payment  is  less  than  the  interest  then  due,  find  the 
amount  of  the  sum  due  to  the  time  when  the  payments  added  together 
shall  he  equal,  at  least,  to  the  interest  already  due ;  then  find  the 
balance,  and  proceed  as  before. 

EXAMPLES, 

tn  which  every  payment  exceeds  the  interest  then  due. 
(33.)    $2,400.  Boston,  Oct.  1st,  1830. 

For  value  received,  I  promise  to  pay  Joseph  Careless, 
or  bearer,  twenty-four  hundred  dollars,  on  demand,  with  interest. 

John  Slack. 
l"i=S  ^If /mWI's'r/f  S:  \     Ti-.  6.O.,  8«o.  15d.,  3„o.  15d. 
Indorsement,  April  1st,  1832,  of  $900.  S 
Settlement,  July  21st,  1833.     What  was  the  balance  due  ? 

Principal  of  the  note, $2,400.00 

Interest,  April  1st,  1831, 72.00 

Amount,  April  1st,  1831,       ....-**        $2,472.00 

Payment,  April  1st,  1831, 200.00 

Balance,  April  1st,  1831, $2,272.00 

Interest,  Dec.  16th,  1831, 96.56 

Amount,  Dec.  16th,  1831, $2,368.56 

Payment,  Dec.  16th,  1831, 300.00 

Q.  What  is  the  General  Rule  when  each  payment  exceeds  the  interest  then 
due?  31.  What,  when  any  payment  is  less  than  the  interest  due  ?  32.  From 
what  source  is  the  rule  derived  ?  30.  When  may  the  former  rule  be  employed  ?  29. 


INTERESf.  I8t 

fialanc6,  Dec.  16th,  1831,  .-.-.--  $2,068.56 

Interest,  April  1st,  1832, 36.20 

Amount,  April  1st,  1832, $2,104.76 

Payment,  April  1st,  1832,         .        ^        ^        .        .        .        .         900.00 

Balance,  April  1st,  1832, $1,204.76 

Interest,  July  21st,  1833,  94.37* 

Balance,  July  21st,  1833, A.  $1,299.13 

34.  On  a  note  of  hand  for  $1,000,  payable  July  1st,  1835,  were 
received  the  following  indorsements,  viz. 

Received,  April  21st,  1836,  $200.  ^      Time.  9,  20,  3,  15,  8,  20,  4,  15,  6, 
Received,  Aug.     6th,  1836,  $150.     20,  1,  3. 

Received,  April  26th,  1837,  $.300.  }     Results.     84833  ;  71318  ;  44408  j 
Received,  Sept.  11th,  1837,  $240.     21407  ;  11  21. 
Received,  April     lst>  1838^  $210.  j  Answer,  $12.05. 

Settlement,  July  1st,  1839. 

EXAMPLES  m 

In  which  every  payment  does  not  exceed  the  interest  then  due. 

35.  On  a  note  given  for  $600,  dated  March  1st,  1822,  witR  inter-- 
est,  there  were  indorsed  the  following  sums. 

May     1st,  1823,  received  $200.  ^ 

June  10th,  1824,  received  $  80. 

Sept.  17th,  1825,  received  $  12.  (      Time— 1,  2,  1,  Ij  15,  1,  8,  15,  1,  7, 

Dec.   19th,  1825,  received  $  15.  (15,  10,  15. 

March  1st,  1826,  received  $100. 

Oct.    10th,  1827,  received  $150.  J 
Settlement,  August  31st,  1828.    What  was  the  balance  due? 

Principal,        -        -        * $600.00 

Interest,  May  1,  1823, 42.00 

Amount,  May  1,  1823, $642.00 

Payment,  May  1,  1823,        ....        ^        .         -        -  200.00 

Balance,  May  1,  1823, *        -  $442.00 

Interest,  June  16,  1824, 29.83 

Amount,  June  16,  1824, $471.83 

Payment,  June  16,  1824,     ....        ^        -        .        -  80.00 

Balance,  June  16,  1824, $391.83 

Interest,  March  1,  1826, 40.16 

Amount,  March  1,  1826,           .......  $431.99 

Payment,  Sept.  17,  1825,     .....-$  12.00 

Payment,  Dec.  19,  1825,  -  ...  $  15.00 

Payment,  March  1,  1826, $100.00  $127.00 

Balance,  March  1,  1826, $304.90 

Interest,  Oct.  16,  1827,        .......        ^  29.74 

Amount,  Oct.  16,  1827,    ....-***  $334.73 

Payment,  Oct.  16,  ]&27,      .......*  150.00 

Balance,  Oct.  16,  1827, $184.73 

Interest,  Aug.  31,  1828, 9.70 

Balance,  Aug.  31,  1828, Answer,  $194.43 

36.  On  a  note  dated  June  16th,  1820,  given  for  $900,  with  inter 
est,  were  indorsed  the  following  payments  : 

*  When  the  mills  are  5  or  more,  add  another  cent;  but  when  less  tban  5,  reject  them 


188 


ARITHMETIC. 


Received,  July  1st,  1821,      $150.  ^ 

Received,  Sept.  ICth,  1822,  $  90.        Time.  1,  15,  1,  2,  15,  2,  11,  1,  6, 
Received,  Dec.  10th,  1824,  $  10.  i  15,  1,  6. 

Received,  June  1st,  1825,     $  20.  (     Results.    80,625;  77,470;  68,027, 
Received,  Aug.  ]6th,  1825,  $200.     44,319.  A.   $483.08. 

Received,  March  1st,  1827,  $300.  J 
Settlement,  Sept.  1st,  1828.     What  was  the  balance  due  ? 

(37. )  $1,600.  For  value  received,  I  promise  to  pay  Rufus  Stanly, 
or  order,  sixteen  hundred  dollars,  with  interest. 

Albany,  July  1st,  1830.  Jonathan  Overton. 

J/tcZorsmen^s.— Received,  Oct.  16th,  1830,  $200.  Jan.  1st,  1831,  $200. 
May  26th,  1831,  $500.  November  1st,  1831,  $15.  February  11th,  1832,  $25. 
June  6th,  1832,  $11.  November  26th,  1832,  $11.  December  1st,  1832,  $5. 
January  11th,  1833,  $24,  and  the  balance  November  26th,  1835.  What  was 
the  balance?     Time.  3,  15,  2,  15,  4,  25,  1,  7,  15,  2,  10,  15. 

Results.  143,206;  125,355;  78,889;  78,763.     A.  $946.14. 

^CONNECTICUT     RULE. 

Established  by  the  Supreme  Court  of  the  State  of  Connecticut  in  1804. 

38.  ."  Compute  the  interest  to  the  time  of  the  first  payment ;  if  that 
be  one  year  or  more  from  the  time  the  interest  commenced,  add  it  to 
the  principal,  and  deduct  the  payment  from  the  sum  total.  If  there  he 
after  payments  made,  compute  the  interest  on  the  lalance  due  to  the 
next  payment,  and  then  deduct  the  payment  as  above ;  and,  in  like 
manner,  from  one  payment  to  another,  till  all  the  payments  are  ab- 
sorbed ;  pvovided  the  time  between  one  payment  and  another  be  one 
year  or  more.  But  if  any  payments  be  made  before  one  yearns  interest 
hath  accrued,  then  compute  the  interest  on  the  principal  sum  due  on 
the  obligation,  for  one  year,  add  it  to  the  principal,  and  compute  the 
interest  on  the  sum  paid,  from  the  time  it  loas  paid  up  to  the  end  of 
the  year ;  add  it  to  the  sum  paid,  and  deduct  that  sum  from  the 
principal  and  interest,  added  as  above.*  " 

"7/"  any  payments  be  made  of  a  less  sum  than  the  interest  arisen  at 
the  time  of  such  payment,  no  interest  is  to  be  computed,  but  only  on 
the  principal  sum  for  any  period.''^ 

39.  For  value  received,  I  promise  to  pay  Peter  Trusty,  or  order, 
one  thousand  dollars,  with  interest.     June  16th,  1824. 

.$1,000.  James  Paywell. 

INDORSEMENTS. 

July    1st,  1825,  received  $250.  ^ 

Aug.  16th,  1826,  received  $157.  lrp-^„    ••    »-    i    i    in   i    a   ir  c  a  in 
Dec.    1st,  1826,  received  $  87.  f^^™^*  1'  ^^'  ^'  ^^  ^^'  ''  ^'  ^^'  ^'^'  ^^^ 
Feb.  16th,  1828,  received  $218.  J 
Settlement,  Aug.  26th,  1828.     What  was  the  balance? 

Q.  How  do  you  dispose  of  the  first  payment  by  the  Connecticut  rule  ?  38. 
What  is  to  be  done  with  the  other  payments  ?  38.  What  exceptions  are  men- 
tioned?  38. 

*  If  a  year  does  not  extend  beyond  the  time  of  payment ;  but  if  it  does,  then  find  the 
amount  of  the  principal  remaining  unpaid,  up  to  the  time  of  settlement,  likewise  tha 
amount  of  the  payment  or  payments  from  the  time  they  were  paid  to  the  time  of  settle 
ment,  and  deduct  the  sum  of  these  several  amounts  from  the  amount  of  the  principal- 


COMPOUND    INTEREST.  189 

Principal  of  the  note,      ...  ....      $1,000.00 

Interest,  July  1,  1825,          -        .                62.50 

Amount,  July  1,  1825,     ........  1,062.50 

Payment,  July  1,  1825,          -         -                ...                .  250.00 

Balance,  July  I,  1825, 812.50 

Interest,  Aug.  16,  1826, 54.84 

Amount,  Aug.  16,  1826, 867.34 

Payment,  Aug.  16,  1826, 157.00 

Balance,  Aug.  16,  1826,  -        - 710.34 

Interest  for  one  year, 42.62 

Amount,  Aug.  16,  1827, 752.96 

Payment,  Dec.  1,  1826, $87.00 

Interest,  Aug.  16,  1827, 3.69  90.69 

Balance,  Aug.  16,  1827,      .        -        - 662.27 

Interest,  Feb.  16,  1828, 19.87 

Amount,  Feb.  16,  1828, 682.14 

Payment,  Feb.  16,  1828, 218.00 

Balance,  Feb.  16,  1828, 464.14 

Interest,  Aug.  26,  1828, 14.70 

Balance,  Aug.  26,  1828, Ans.  $478.84. 

(40.)  $875.  For  value  received,  I  promise  to  pay  Daniel  Bur 
gess,  or  order,  eight  hundred  and  seventy-five  dollars,  with  interest. 

Hartford,  January  10th,  1821.  Henry  Frothing. 

Irtdorsemcnts.— Received  $260,  August  10th,  1824.  $300,  December  16th, 
1825.  $50,  March  1st,  1826.  $150,  July  1st,  1827.  What  was  there  due 
September  1st,  1828?  Time— 3,  7,  1,  4,  6,  1,  91,  6, 15, 1,  2.  Results— 80,313 ; 
56,818;  54,990;  41,777.  "  A.  $447.01. 


COMPOUND    INTEREST. 

LXXXH.  1.  Compound  Interest  is  the  premium  given  for  the 
use  of  both  the  principal  and  its  interest  when  the  latter  becomes  due 
and  remains  unpaid.     This  is  sometimes  called  interest  upon  interest. 

2.  Simple  interest  implies,  as  we  have  seen,  (lxxx.  3,)  that  the 
interest  is  payable  annually ;  hence,  to  jfind  the  compound  interest, 
we  may  proceed  as  follows :  * 

RULE. 

3.  Find  the  amount  of  the  principal  for  one  year,{unless  a  different 
time  he  named,)  then  of  this  amount  as  before,  and  so  on  to  the  time 
of  settlement. 

4.  Subtract  the  given  sum  from  the  last  amount,  and  the  remainder 
ivill  be  the  compound  interest  required. 

LXXXII.  Q.  What  is  Compound  Interest?  1.  What  is  the  rule  for  finding 
the  amount?  3.  What,  for  finding  the  compound  interest?  4.  Why  should  the 
intei*est  be  compounded  annually  ?  2. 

*  Compound  interest,  though  just,  is  not  legal.   • 


190  ARITHMETIC. 

5.  What  is  the  compound  interest  of  $156,  for  2  years,  and  what 
is  the  amount  ] 

$  1  5  C=given  sum  or  first  principal. 
6=rate  per  cent,  understood. 
9 . 3  6=interest  for  the  first  year. 

15  6        =  principal  for  the  first  year. 

16  5.3  6=amount,  principal  the  second  year. 

6=rate  per  cent,  understood. 
9.9  2  1  6=  interest  for  the  second  year. 
1  6  5.3  6        ^principal  for  the  second  year. 
A.     $175. 281  6=  amount  for  two  years. 
15  6  =given  sum  deducted. 

A.     8  19.281  6=  compound  interest  for  two  years. 

6.  What  is  the  compound  interest  of  $500  for  4  years  % 

A.  $131.238yV+. 

7.  What  is  the  compound  interest  of  $15,000  for  5  years  at  7  per 
cent.?  A.  $6038.275yV 

8.  What  is  the  amount  of  $13,000  for  3  years  at  compound  inter- 
est, the  rate  being  4|^  per  cent.  1  A.  $14835.159. 

9.  What  will  $600  amount  to  at  compound  interest  in  4  years  at 
7  per  cent.,  the  interest  being  payable  semi-annually?  Find  the 
amount  of  $600  for  6  months ;  then  of  this  amount  for  another  6 
months ;  and  so  on  for  the  whole  time.  A.  $790,079. 

10.  What  will  be  the  compound  interest  of  $140  for  3  years,  it 
being  payable  semi-annually  ?  A.  $27.16. 

11.  What  is  the  compound  interest  of  $240,  payable  quarterly,  for 
2  years,  at  7  per  cent.?  A.  $35,728. 

12.  What  is  the  compound  interest  of  $1,000  for  2  years  at  3^  per 
per  cent,  payable  quarterly  ?  A.  $72.18. 

13.  What  is  the  compound  interest  of  $750  for  5  years  and  6 
months,  payable  annually  ?  Find  the  amount  for  5  years,  then  for  6 
months.  .4.  $283.78. 

14.  What  is  the  amount  at  compound  interest  of  $300  at  7  per 
cent,  for  3  years  4  months  and  15  days?  A.  $377.15. 

15.  If  a  note  of  $00.60,  dated  October  25th,  1830,  with  the  interest 
payable  annually,  be  paid  October  25th,  1840,  what  will  it  amount  to 
at  compound  interest  ?  A.  $76.51. 

16.  Find  the  balance  due  on  the  following  note,  (by  lxxxi.  31,  32,) 
compounding  the  interest  annually. 

$1,000.  On  demand,  for  value  received,  I  promise  to  pay  John 
Stearns,  or  order,  one  thousand  dollars,  with  interest. 

Joseph  Discount. 

Hartford,  August  1st,  1830. 

This  note  has  $500  indorsed  on  the  back  of  it  January  16th,  1836,  and  was 
paid  in  full  February  1st,  1840.  .  A.  $1107.46. 


COMPOUND    INTEREST. 


191 


17.  If  the  number  of  colored  persons  in  the  United  States  at  the 
present  time  (1840)  be,  as  is  supposed  by  some,  three  millions,  and 
their  rate  of  increase  25  per  cent,  in  ten  years,  what  will  be  their 
number  in  18601— in  1900?  A.  4,087,500;  11,444,090. 

18.  As  ^2  at  compound  interest  amounts  to  2  times  as  much  as  61 ; 
$3,  3  times  as  much,  and  so  on,  we  may  make  a  table  containing  the 
amount  of  £1  or  $1  for  several  years,  by  which  the  amount  of  any 
sum  may  be  easily  found  by  simply  multiplying  once. 

TABLE, 

Showing  the  amount  of  £1  or  $l,for  30  years  at  5,  G,  and 7 per  cent, 
compound  interest. 


Years. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

1 

1.050000 

1.060000 

1.070000 

2 

1.10  2  5  0  0 

1.12  3  6  0  0 

1.14  4  9  0  0 

3 

1.15  7  6  2  5 

1.19  10  16 

1.225043 

4 

1.215506 

1.262477 

1.310795 

5 

1.27628  1 

1.338225 

1.402552 

6 

1.340095 

1.418519 

1.500730 

7 

1.407100 

1.503630 

1.605781 

8 

1.477455 

1.593848 

1.718186 

9 

1.551328 

1.689479 

1.838459 

10 

1.628894 

1.790848 

1.967151 

11 

1.710339 

1.898299 

2.104852 

12 

1.795856 

2.012197 

2.252191 

13 

1.885649 

2.132928 

2.409845 

14 

1.979931 

2.260904 

2.578534 

15 

2.078928 

2.396558 

2.759032 

16 

2.182875 

2.540352 

2.952164 

17 

2.292018 

2.692773 

3.158815 

18 

2.406619 

2.854339 

3.379932 

19 

2.526950 

3.025600 

3.616528 

20 

2.653297 

3.207136 

3.869685 

21 

2.785963 

3.399564 

4.140563 

22 

2.925260 

3.603539 

4.430403 

23 

3  .  0  7  1  ,5  2  4 

3.819750 

4.740530 

24 

3.225100 

4.048935 

5.072367 

25 

3.386355 

4.291870 

5.427434 

26 

3.555673 

4.549383 

6.807352 

27 

3.733457 

4.822347 

6.213868 

28 

3.920130 

5.111688 

6.648838 

29 

4.1  16  13  6 

5.418389 

7.1  14257 

30 

4.321943 

5.743493 

7.612255 

19.  What  is  the  compound  interest  of  $20.15  for  4  years  at  6  per 
cent.  1  By  the  Table  the  amount  of  $1  for  4  years  is  $1.262477  X 
$20.15=$25.438+frora  which  subtracting  $20.15  leaves  5.288y\.+ 

A.  S5.288y^9.+ 

20.  What  is  the  compound  interest  of  $2,000  for  ten  years  at  7 
per  cent.  ?  A.  $1,934.30.  At  6  per  cent. "?  A.  $1,581,696.  At 
5  per  cent.?     A.  $1,257.79. 


193  ARITHMETIC, 

21.  What  is  the  compound  interest  of  S300  for  20  years  at  7  per 
cent?     ^.$860.90.         At  G  per  cent.  1     A.  $662.14. 

22.  What  is  the  amount  of  $600  for  30  years  at  6  per  cent,  com- 
pound interest]     A.  $3,446.10.        For  15Y.  6mo.  1  [See  13.] 

A,  $1,481.07. 

23.  To  what  sum  will  $500  amount  in  17  years  4  months  and  15 
days  at  compound  interest  ?  A.  $1,376.68. 

24.  What  is  the  amount  of  $200  for  45  years,  at  7  per  cent,  com- 
pound interest]  As  45  years  extend  beyond  the  Table,  find  the  amount 
for  any  number  of  years  in  it  at  first,  say  20  years,  then  of  this  amount 
for  20  more;  finally  for  the  remaining  5  years.     A.  $4,200.49. 

25.  What  is  the  amount  of  $6,000  for  60  years,  the  compound  in- 
terest being  at  the  rate  of  7  per  cent.  1  A.  $347,678.56. 

26.  What  is  the  amount  of  $600  for  11  years  10  months  and  23 
days,  at  6  per  cent.  ]  the  interest  compounded  annually  1 

A.  $1,200.294yV 

27.  What  is  the  amount  of  $600  for  16  years  8  months  at  6  per 
cent,  simple  interest  ?  A.  $1,200. 

Note. — By  the  last  two  examples,  it  appears  that  any  sum  at  6 
per  cent,  compound  interest,  will  double  in  11  years  10  months  and 
from  22  to  23  days,  while  at  simple  interest  it  would  require  16  years 
and  8  months.* 


DISCOUNT. 

LXXXIII.  1.  Discount  is  that  deduction  which  is  made  for 
paying  money  before  it  is  due. 

2.  Present  worth  of  any  sum  implies  that  it  is  payable  at  a  future 
day  without  interest. 

3.  The  PRESENT  worth,  then,  is  and  ought  to  be  such  a  sum  as 
would  at  interest  amount  to  the  debt  when  due. 

4.  Thus  the  present  worth  of  $106,  due  1  year  hence,  is  $100,  and 
the  (Uscount  $6  ;  for  $100  at  interest  for  that  time  amounts  to  $106. 

5.  The  discount  of  any  sum  is  less  than  its  interest ;  thus  the  dis- 

LXXXIII.  Q.  What  is  Discount?  1.  Present  worth?  2.  What  does  it 
imply  ?  3.  What  is  the  present  worth  and  what  the  discount  of  $106,  due  1 
year  hence?  4.  What  is  the  interest  of  $106  for  1  year?  5.  Which  then  is 
the  most,  the  interest  or  the  discount  ?  5. 

*  It  seems  there  is  considerable  difference  between  simple  and  compound  interest  even 
for  a  short  time,  and  when  the  latter  is  permitted  to  accumulate  for  ages  it  amounts  to 
a  sum  almost  incredible.  For  example,  suppose  a  cent  had  been  put  at  interest  at  the 
commencement  of  the  Christian  era,  it  would  have  amounted  at  the  end  of  the  year 
1827,  to  only  $l,106-n7.  But  the  compound  interest  of  the  same  sum  for  the  same  time 
would  have  amounted  to  a  sum  greater  than  can  be  contained  in  6,000,000  of  globes, 
each  equal  to  our  earth  in  magnitude  and  all  of  solid  gold;  or  to  $172,616,474,047,552,- 
529, 470,760,914,974,711,959,976,620,354toPo  nearly. 


DISCOUNT.  193 

count  of  $106  for  a  year  is  $6,  but  the  interest  of  $106  for  that  time 
is  $6,36. 

6.  The  debt  then  may  be  regarded  as  the  amount^  the  present 
worth  as  the  principal,  and  the  discount  as  the  interest  of  this  prin- 
cipal but  not  of  the  debt. 

7.  Hence  finding  the  present  worth  is  the  same  process  in  effect  as 
that  for  finding  the  principal  in  Interest,  Case  xi.,  which  maybe  ex- 
pressed thus, — 

RULE. 

8.  Divide  the  given  sum  or  debt  by  the  amount  o/  $1  for  the  given 
time. 

9.  The  quotient  will  represent  the  present  worth,  which  taken  from 
the  debt  will  leave  the  discount. 

10.  What  is  the  present  worth  of  $133.20,  payable  1  year  and  10 
months  hence,  and  what  the  discount  1 

Note.— The  amount  of  $1  for  lY.  10m.  is  Si. 11,  then  $133.20-^ 
$l.ll=$120=for  the  present  worth,  and  $120  from  $133.20  leaves 
$13.20  for  the  discount.  A.  $120;  $13.20. 

11.  For  the  proof,  find  the  interest  of  $120  for  1  year  lOmo.  then 
its  amount ;  and  if  it  make  $133.20  the  work  is  right. 

12.  What  is  the  present  worth  of  $660  due  1  year  and  8  months 
hence  ?    What  its  discount  ?  A.  $600 ;  $60. 

13.  What  sum  of  ready  money  is  equivalent  to  $460  due  2  years 
and  6  months  hence.     Wliat  sum  is  equal  to  the  discount  % 

A.  $400;  $60. 

14.  If  I  pay  a  debt  of  $1,350,  5  years  and  10  months  before  it  is 
due,  what  sum  ought  I  to  pay  and  what  discount  ought  to  be  made 
meT  A.  $1000;  $350. 

15.  Suppose  you  have  owing  to  you  $3065. 62|  payable  in  2  years 
8  months  15  days,  and  money  is  worth  no  more  than  5  per  cent.  ; 
what  sum  of  ready  money  can  you  afford  to  take,  and  what  will  the 
discount  amount  to  ?  A.  $2,700;  $365.62^. 

16.  Whatsis  the  difference  in  value  between  $699.25  cash,  and 
$751,116,  due  lY.  6m.  hence,  when  money  is  worth  only  4  per  cent.l 

A.  $9.35. 

17.  If  I  am  offered  goods  for  $2,500  cash,  or  for  $2,821.50  on  "  9 
months  ;"  which  is  the  best  offer,  and  by  how  much "? 

A.  Cash  by  $200. 

18.  Suppose  a  merchant  contracts  a  debt  of  $24,000,  to  be  paid  in 
four  installments,  as  follows,  viz  :  one  fifth  in  4  months  ;  one  quar- 
ter in  9  months  ;  one  sixth  in  one  year  and  2  months,  and  the  rest  in 
1  year  and  7  months ;  what  is  the  present  worth  of  the  whole  sum  ? 

A.  $22,587.651. 

Q.  What  terms  in  Discount  resemble  those  in  Interest  ?  6.  Which  operation 
in  the  one  is  the  same  in  effect  as  in  the  other  ?  7.  Rule  ?  8.  What  is  the 
discount  of  $104  for  4  months?— of  $208  for  8  months?— of  $109  for  1  year  6 
months  ? 

17 


194  ARITHMETIC. 

19.  Suppose  I  contract  to  receive  flour  at  different  times,  from 
New-York,  on  9  months'  credit,  and  receive  as  follovi'S,  viz  : 

Jan.    16,1830,   180  barrels  at  S 10    per  barrel.  $ 

Feb.    20,  1830,  900  barrels  at  $9^  per  barrel.  $ 

April  16,  1830,  850  barrels  at  $lo|  per  barrel.  $ 

June  21,  1830,  600  barrels  at  $11  per  barrel.  S 

Oct.     10,  1830,  950  barrels  at  $10^  per  barrel.  $- 


Now  suppose  I  remit  the  cash  in  payment  as  often  as  I  receive  a 
lot  of  flour,  what  ought  to  be  the  sum  total  of  all  my  remittances, 
when  money  (being  "  tight")  is  worth  at  least  10  per  cent.  1 

A.  $33,151,162. 


DISCOUNT  BY    COMPOUND   INTEREST. 

RULE. 
LXXXIV.     1.  Divide  the  given  sum  hy  the  amount  o/  $1  at  com- 
pound interest  for  the  stated  time  ;  the  quotient  will  be  the  present 
worth,  which,  subtracted  from  the  given  sum,  will  leave  the  discount. 

2.  For  the  quotient,  which  is  the  present  worth ;  multiplied  by 
the  divisor,  which  is  the  amount  of  $  1  for  the  whole  time ;  must  re- 
produce the  dividend,  which  is  the  given  sum  or  amount. 

3.  What  is  the  present  worth  of  $561.80,  due  2  years  hence,  reck- 
oning 6  per  cent,  per  annum,  compound  interest?  A.  $500. 

4.  What  sum  in  cash  is  equivalent  to  8087,512  iW,  payable  2  years 
hence,  deducting  7  per  cent.,  compound  interest  1        A.  $600.50. 

5.  How  much  discount  for  the  cash  ought  to  be  made  on 
$2,127.778fo,  due  6  years  hence,  reckoning  compound  interest 
yearly?  A.  $627.778pV 

6.  Suppose  I  propose  to  sell  you  the  following  note  at  a  discount  of 
7  per  cent,  per  annum,  compound  interest ;  what  sum  do  I  ask  for  it  ? 

$11,792.47.  New  York,  April  1st,  1828. 

For  value  received,  I  promise  to  pay  on  the  first  day  of 
September,  eighteen  hundred  and  thirty,  unto  Peter  Hunks,  or  order, 
eleven  thousand  seven  hundred  and  ninety-two  yVo  dollars. 

William  Neverfail. 
A.  $10,008.09. 

7.  Suppose  a  father's  estate  was  so  divided  between  two  sons,  one 
20  years  (Id  and  the  other  only  one  year  old,  that  each  on  arriving 
"  at  age"  should  receive  an  equal  portion.  Suppose,  also,  that  when 
the  younger  brother  was  21,  the  older  brother's  portion,  by  means  of 
annual  loans  at  compound  interest  amounted  to  $3,207,136,  how 
many  dollars  was  each  to  receive  when  21  years  old  ?     A.  $1,060. 

LXXXIV.  Q.  What  is  the  rule  for  finding  the  discount  when  the  interest 
has  been  compounded  ?  1 .    What  is  the  reason  for  it  ?  2. 


BANKING.  195 


BANKING 

.  LXXXV.  1.  A  Bank  is  an  incorporated  institution,  that  deals  in 
money.  Its  capital,  which  is  limited  by  law,  is  usually  owned  in 
shares  by  persons  called  Stockholders. 

2.  The  proper  business  of  a  bank  is  to  make  and  lend  notes  called 
"  hank  bills,''''  which  circulate  as  money,  because  the  bank  is  obliged 
to  redeem  them  with  specie. 

3.  When  the  banks  loan  money,  it  is  their  custom  to  take  the  in- 
terest in  advance  ;  that  is,  to  deduct  it  from  the  face  of  the  note  at 
the  time  the  money  is  lent.  The  note  is  thence  said  to  be  dis- 
counted. 

4.  The  face  of  every  note,  therefore,  should  exceed  the  sum  re- 
ceived or  wanted,  as  much  as  will  just  equal  the  interest  of  the  note 
to  the  time  when  it  is  payable 

5.  Hence  the  sum  discounted  is  called  the  Amount  ;  the  interest 
deducted  the  Discount,  and  the  remainder  the  proceeds,  or  more  cor- 
rectly, the  Present  Worth. 

C.  A  note  to  be  discounted  or  bankable,  must  be  made  payable  at 
a  future  day,  and  to  the  order  of  some  person  who  indorses  it. 

7.  The  indorser,  however,  is  not  responsible  for  its  payment  unless 
notified  that  the  note  is  due  and  demanded,  but  not  paid.* 

8.  The  banks  take  interest  for  3  days  more  than  the  time  specified 
in  the  note,  because  the  debtor  is  not  obliged  by  law  to  make  payment 
till  the  same  3  days  have  elapsed,  which  are  thence  called  days  of 
grace. 

RULE. 

9.  Cast  the  interest  on  the  note  for  3  days  more  than  the  time 
specified;  then  deduct  the  interest  from  the  face  of  the  note,  and  the 
remainder  will  be  the  sum  loaned. 

10.  What  is  the  bank  discount  on  $600,  payable  in  60  days  1  Th^ 
interest  of  $600  for  [60d.+3d.=]  63d. =$6.30.  A.  $630. 

11.  What  is  the  bank  discount  on  $1,200,  payable  90  days  hence, 
and  what  would  be  its  present  worth  ]         A.  $18.60  ;  $1,181.40. 

12.  $1,800.     Sixty  days  after  date,  for  value  received,  I  promise 

LXXXV.  Q.  What  is  a  Bank?  1.  What  is  said  of  its  capital?  1.  What, 
of  its  business?  2.  When  is  a  note  said  to  be  discounted?  3.  When  a  partic- 
ular sum  is  wanted  at  bank,  what  sum  should  be  named  in  the  note?  4. 
What  are  meant  by  bank  Discount,  Amount,  and  Present  Worth  ?  5.     What 

Particulars  must  be  observed  in  writing  a  bankable  note  ?  6.  What  is  meant 
y  a  protest  ?  7.  Wliat  by  3  days  of  grace  ?  8.  What  is  the  rule  for  ascertain- 
ing the  sum  loaned  or  received  ?  9.  What  sum  would  be  received  on  a  note  of 
$60  for  2  months  ?— for  4  months  ?— for  6  months  ?— for  8  months  ?— for  10 
months  ? 

*  A  legal  notice  in  writing  is  called  a  protest. 


196  ARITHMETIC. 

to  pay  Peter  Parley,  or  order,  at  the  Etna  Bank  (N.  York,)  eighteen 
hundred  dollars.  Peter  Paywell. 

Suppose  "  old  Mr.  Peter  Parley  "  indorses  the  above  note,  and  it  is 
discounted,  what  sum  would  Mr.  Paywell  receive?   A.  Sl,777.95. 

13.  What  sum  would  be  the  present  worth  of  $1,200  discounted  at 
bank  and  payable  in  60  days,  at  7  per  cent.  ?  A.  $1,185.30. 

14.  A  merchant  sold  250  bales  of  cotton,  each  weighing  300  pounds, 
for  12|-  cents  per  lb.  which  cost  him  the  same  day  10  cents  per  lb. ; 
he  received  in  payment  good  paper  for  4  months  time.  Now  suppos- 
ing he  gets  this  note  discounted  at  bank,  what  will  be  his  profits  I 

A.  $1,682.81  i^oV 

15.  To  find  what  sum  or  amount  must  be  named  in  a  note  in  order 
to  obtain  a  particular  loan  at  bank. 

RULE. 

16.  Deduct  the  hank  discount  on  $1  /or  the  given  time  from  $1,  and 
divide  the  desired  loan  by  the  remainder,  the  quotient  will  he  the  sum 
or  amount  required. 

17.  For  if  the  quotient,  which  is  the  required  amount,  be  multiplied 
by  the  divisor  which  is  the  present  worth  of  $1,  for  the  given  time, 
the  process  must  re-produce  the  dividend,  which  is  the  given  loan. 

18.  Most  paper  at  our  banks  is  discounted  either  for  95  days  or  4 
months.  The  interest  of  $1  for  3  days  (grace)  is  $.0005,  and  for 
95d.+3d.=$.01633  nearly;  for  4mo.+3d.==$.0205 ;  then  $1- 
$.01633=$.98367  ;  and  $l-$.0205=$.9705,  therefore  : 

19.  The  divisor  under  this  rule  for  any  note  payable  in  95  days^  is 
$.98367,  and  for  4  months,  $.9795. 

20.  Suppose  I  want  a  loan  at  bank  of  $14,842.50  for  60  days : 
what  sum  must  be  named  in  the  note  to  obtain  that  amount  of  money  % 
The  interest  of  $1  for  63  days  (  =  .0105)  deducted  from  $1  leaves 
$.9895,  for  a  divisor.  A.  $15,000. 

21.  Suppose  your  note  for  6  months  is  discoiinted  at  bank,  and 
$484.75  passed  to  your  credit ;  what  must  have  been  the  face  of  the 
note?  A.  $500. 

22.  If  I  want  from  a  bank  at  Rochester,  New  York,  $5,786.50 
for  my  note  at  6  months,  what  must  be  the  face  of  the  note  1 

A.  $6,000. 

23.  Suppose  "  old  Mr.  Peter  Parley  "  wants  a  loan  himself  at  bank 
of  $994.50  for  30  days,  at  which  time  he  expects  to  be  able  to  re- 
fund it  from  the  profits  of  his  story  books,  and  that  Mr.  Paywell 
reciprocates  the  favor  shown  to  him  above,  by  indorsing  it ;  what 
sum  must  be  specified  in  the  note  to  obtain  that  loan  1     A.  $1000. 


EQUATION    OF    PAYMENTS.  197 


EQUATION^  OF  PAYMENTS. 

LXXXYI.  1.  In  how  many  months  will  1  dollar  gain  as  much 
interest  as  2  dollars  will  gain  in  6  months  1   A.  6x2  =  12  months. 

2.  In  how  long  time  will  1  dollar  gain  as  much  interest  as  5  dollars 
will  gain  in  12  months'?  A.  60  months. 

3.  How  many  months  is  the  use  of  1  dollar  equivalent'  to  the  use  of 
10  dollars  for  20  months  1  A.  200  months. 

4.  How  long  ought  you  to  lend  B  1  dollar  to  repay  him  for  his  kind- 
ness in  lending  you  100  dollars  for  4  months  1 

A.  400  months  =  33^  years. 

5.  In  what  time  will  the  use  of  100  dollars  be  equivalent  to  the  use 
of  300  dollars  6  months  1  300  dollars  for  6  months  is  the  same  as 
1  dollar  for  [300x6=]  1800  months,  and  100  dollars  is  the  same  of 
course  as  jl^  of  1800  months  ;  that  is,  1800-^  100  =  18mo. 

A.  lY.  6mo. 

6.  A  having  lent  B  200  dollars  for  9  months,  wishes  a  like  favor 
of  B,  but  needs  only  50  dollars ;  how  long  may  A  keep  the  50  dollars 
without  doing  any  injustice  to  B  1  ^.3  years. 

7.  Suppose  A  lends  B  8  dollars  to  be  paid  in  2  months,  and  12 
dollars  to  B.  paid  in  7  months,  making  in  all  20  dollars  lent  B.  Now 
how  long  ought  B  to  lend  A  1  dollar  to  repay  him  for  his  kindness  1 
How  long  ought  B  to  lend  A  20  dollars  ] 

2X   8  =  16  ;  therefore    $8  for  2  mo.  =  $l  for  16  months. 

7 X  12  =  84  ;  therefore  $12  for  7  mo.  =$1  for  84  months. 

$20  )100(5mo.  A 

8.  Then  B  ought  to  lend  A  1  dollar  100  months,  but  20  dollars 
only  aV  as  long;  that  is,  100^20=5  months.  Therefore  20  dollars 
payable  in  5  months  is  the  same  as  if  8  dollars  of  the  $20  were  pay- 
able in  2  months,  and  the  remaining  12  dollars  in  7  months. 

9.  Proof — The  interest  of   $8  for  2  months  is     .     .       8  cents. 

The  interest  of  $12  for  7  months  is    .     .     42  cents. 
The  interest  of  S20  for  5  months  is    .     .     50  cents. 
RULE. 

10.  Multiply  each  payment  by  the  time,  and  divide  the  sum  of  these 

LXXXVI.  Q.  In  how  many  months  will  1  dollar  gain  as  much  interest  as  6 
dollars  in  3  months  ? — as  6  dollars  in  4  months  ? — in  8  months  ?  Plow  long 
ought  A  to  lend  me  12  dollars  to  reciprocate  ^  my  favor  in  lending  him  6 
dollars  for  2  months  ?  How  long  ought  I  to  lend  you  20  dollars  to  recompense 
you  for  lending  me  onetime  8  dollars  for  2  months,  and  at  another  time  12  dollars 
for  7  months?   [See  7.]    Why  divide  by  20  dollars  ?  8.     What  is  the  rule  ?  10. 

1  Equation,  [L.  ccquatio.']    Literally,  a  making  equal,  or  an  equal  division. 

2.  Eqvivalent.  Equal  in  value  or  worth;  equal  in  force,  power,  or  effect.  Of 
the  same  import  or  meaning. 

1.  Reciprocate,  [L.  recipricu3.'\  To  exchange  ;  to  interchange;  to  give  and  return 
mutually. 

17* 


198  ARITHMETIC. 

several  products  by  the  sum  of  the  payments;  the  quotient  will  be  the 
mean  or  equitable  time  for  the  'payjnent  of  the  ivhole* 

11.  A  owes  B  $200  to  be  paid  in  6  months,  $300  in  12  months, 
$500  in  3  months  ;  what  is  the  equated  time  for  the  payment  of  the 
whole?  A.  6yV 

12.  What  is  the  equated  time  for  paying  S2,000,  of  which  S500 
is  due  in  3  months,  $360  in  5  months,  and  $600  in  8  months,  and 
the  balance  in  9  months  T  A.  Q^iw^—^^^  mo. 

13.  A  merchant  bought  goods  amounting  to  $1,200,  |  of  which  he 
was  to  pay  in  cash,  ^  in  6  months,  and  the  balance  in  10  months; 
what  was  the  equitable  time  for  the  payment  of  the  whole  ? 

A.  6g-  months. 

14.  A  merchant  proposed  to  sell  goods  amounting  to  84,000  on  8 
months  credit ;  but  the  purchaser  preferred  to  pay  \  in  cash  and  \  in 
3  months ;  what  time  should  be  allowed  him  for  the  payment  of  the 
remainder  '?  A.  2Y.  5m. 

15.  A  having  sold  B  a  bill  of  goods  amounting  to  $1,200,  left  it 
optional  with  him  either  to  take  them  on  8  months^  credit,  or  to  pay 
^  in  cash,  ^  in  two  months,  ^  in  4  months,  and  the  remainder  at  an 
equated  time  for  paying  the  balance  on  the  terms  first  named.  What 
was  that  time  ]  A.  4Y.  4m. 

*  This  rule  proceeds  on  the  supposition,  that  what  is  gained  by  keeping  the  money 
after  it  is  due  is  equal  to  what  is  lost  by  paying  it  before  it  is  due.  But  this  is  not  ex- 
actly true,  for  the  gain  is  equal  to  the  interest,  while  the  loss  is  equal  only  to  the  dis- 
count, which  is  always  less  than  the  interest.  However,  the  error  is  so  trifling,  in 
most  cases  which  occur  in  business,  as  not  to  make  any  material  difference  in  the 
resttlt. 


SIMPLE  PROPORTION/ 

OR 

THE  RULE  OF  THREE, 

SOMETIMES  CALLED  THE  GOLDEN  RULE. 


BY     ANALYSIS.- 

LXXXVII.   1.  If  1  hat  costs  ^5,  what  wUl  4  hats  cost  T  A.  $20. 

2.  If  1  quarter  of  a  yard  of  hlue  satin  costs  37|  cents,  what  will  1 
yard  cost?    What  will  315yd.  3qr.  cost  T       ^.  $1.50  ;  $473.62|. 

3.  If  1  pound  of  sugar  will  cost  9f  cents,  what  will  be  the  cost  of 
Icwt.  ■?— of  Icwt.  3qr.1  ^.$9.75.     $17,062^. 

4.  If  SI.  125  will  buy  1  gallon  of  wine,  how  many  hogsheads  may 
be  bought  for  $70,875  ]  ^.1  hogshead. 

6.  If  Icwt,  3qr.  of  sugar  cost  $17.50,  what  will  1  quarter  cost? 
A.  $2.50.     What  will  1  pound  cost  ?  A.  10  cents, 

6.  If  6  bushels  of  wheat  cost  $12,  what  will  1  bushel  cost  ] — 5 
bushels  cost]— 115  bushels  cost]  A.  $2;  $10;  $230. 

7.  If  400  barrels  of  flour  cost  $4000,  what  will  89  barrels  cost  ? 
Find  the  price  of  1  barrel  first.  A.  $890. 

8.  If  a  farm  consisting  of  300  acres  sells  for  $6,150,  what  would 
a  small  farm  of  50  acres  sell  for  at  that  rate?  A.  $1,025. 

9.  When  10  yards  of  cotton  cloth  clost  $1.50,  what  will  be  the  cost 
of  10  pieces,  each  containing  520-  yards  )  A.  $78.75. 

10.  When  tea  is  £5.  169.  by  the  cwt.,  what  will  Iqr.  of  a  cwt.  cost  ? 
What  will  30  chests,  each  weighing  lOcwt.  Iqr.  cost"? 

^.£1.  9s.;  jE:1,189. 

11.  If  6  ounces  of  silver  will  make  15  »poons,  how  many  spoona 
can  be  made  from  8  silver  tankards;  each  weighing  21bs.  6oz. 

A.  600  =  50doz. 

12.  If  50  dozen  silver  spoons  are  made  from  8  silver  tankards, 
each  weighing  21b.  6oz.,  how  much  silver  will  be  required  to  make  1{ 
dozen,  or  15  spoons'?  A.  6  ounces. 

13.  If  it  require  $300  to  ^ain  $15  interest  in  a  year,  how  much 
will  be  required  to  gain  $100.  A.  $2000. 

14.  If  10  men  will  mow  a  certain  meadow  in  13  days,  how  long  a 
time  will  be  required  for  25  men  to  do  the  samel  1  man  will  be  10 
times  longer  than  10  men.  A.  5i  days. 

LXXXVII.  Q,  When  5  poands  of  cheese  cost  60  cents,  what  will  II  pounds 
cost?  What  will  5^  pounds  of  sugar  cost  at  30  cents  for  3  pounds  ?  Whea 
4  gallons  of  wine  cost  $5,  what  is  the  prrice  of  a  single  quart? 

1,  Proportion,  [L.  proporiio.]  The  comparative  relation  of  any  one  thing  to 
another ;  symmetry  ;  equal  or  just  share  ;  form  ;  size. 

2.  Analysis,  [G.  analusis.}  The  separation  of  a  compound  into  the  parts  that  com- 
pose it ;  a  resolving  a  consideration  of  any  tiling  in  its  separate  parts  ;  it  is  opposed  to 
synthesis,  [G.  suntkesis,']  which  means  the  putting  of  two  or  more  things  together. 
Analysis  in  Arithmetic  is  finding  the  whole  by  first  finding  the  value  of  unity. 


200  ARITHMETIC. 

15.  If  100  men  can  complete  a  job  of  work  in  25  days,  in  how 
many  days  will  7  men  do  the  same  1  A.  357|. 

16.  If  100  men  can  do  a  job  of  work  in  25  days,  how  many  men 
will  be  required  to  do  the  same  in  10  days  1  The  less  days  the  more 
men  will  be  required,  A.  250  men. 

17.  If  the  interest  of  a  certain  sum  is  $10  for  3  years,  what  is  the 
interest  of  the  same  sum  for  12  years  1  A.  $40. 

18.  If  2,400  men  can  do  a  job  of  work  in  6  months,  how  many 
men,  working  at  the  same  rate,  would  do  the  same  job  in  4  months  1 

A.  3,600. 

19.  If  461  bottles  will  hold  5hhd.  30gal.  3qt.  of  cider,  how  many 
hogsheads  will  1161  such  bottles  holdl 

A.  3453qt.  =  13hhd.  44gal.  Iqt. 

20.  If  I  of  a  barrel  of  flour  cost  $4.80,  what  will  ^  of  a  barrel 
cost?     A.  $2.40.     What  will  1  barrel  cost  1  A.  $7.20. 

21.  If  I  of  a  load  of  hay  cost  $10,  what  will  5135-  loads  cost  1 

A.  $6,159. 

22.  If  f  of  a  yard  of  cloth  cost  $6,  what  is  it  a  yard  1  If  §  be  $6, 
then  i  is  }  of  $6  (=2)  and  f  are  -|  of  $6  ==$10.— [See  lii.  10.] 

A.  $10. 

23.  If  ^  of  a  barrel  of  cider  is  sufficient  for  a  family  9  weeks,  how 
long  will  1  barrel  last  them] — 1  barrel  will  last  them  |  of  9  weeks. 
How  long  would  50  barrels  last  the  same  family  T  A.  50  times  21  = 
1050w.-^52w.=20^^  years. 

24.  If  2^  of  a  hogshead  of  molasses  cost  $2,  what  will  120|hhd. 
cost]     Ihhd.  cost  2^5  of$2.  A.  $2011^. 

25.  If  f  of  a  yard  of  broadcloth  cost  $2.40,  what  will  |  cost  ?  1yd. 
cost  I  of '$2.40==$0,  then  |yd.  is  |  of  $6.  A.  $2.25. 

26.  When  j\  of  a  ship  is  valued  at  $20,000,  what  is  f  of  it  worth  T 

A.  $33,000. 

27.  When  f  of  a  gallon  of  oil  costs  |  of  a  dollar,  how  much  will  1 
gallon  cost  ?  How  much  is  |  of  ^  ?  A.  $1-  What  will  40  gallons 
cost?  A.  $35. 

28.  Wlien  I  of  a  dollar  will  buy  §  of  a  bushel  of  corn,  how  much 
will  $200  buy?  A.  137ibu. 

29.  If  I  of  a  pound  of  cassia  cost  f-  of  a  dollar,  what  will  f  of  a 
pound  cost  ?     lib.  cost  f  of  $  J  ;  and  fib  cost  f  of  f  of  $^ 

A.  $f =80  cents. 

30.  If  I  of  a  yard  of  cloth  cost  |  of  a  dollar,  what  will  y\  of  a  yard 
cost?    ^.$.255.     What  will  40f  yards  cost ?  ^.$19,061. 

31.  If  I  of  f  of  a  cask  of  lime  cost  |  of  a  dollar,  what  will  ?  of  j  of 
a  cask  cost  ?     1  cask  costs  |  of  $1.  A.  $1.31^V 

Q.  If  10  men  can  perform  a  job  of  work  in  5  days,  how  many  men  would  be 
required  to  do  the  same  in  10  days?— in  20  days  ?  [See  16.]  When 4  bushels 
of  rye  cost  $3,  what  will  f  of  that  quantity  cost  ?  If  f  of  a  dollar  will  buy  5 
yards  of  calico,  what  ought  f  of  a  yard  to  cost  ?  When  |  of  a  barrel  of  flour 
sells  for  $6,  why  is  f  of  $6  the  price  of  1  yard?  22.  If  |  of  a  yard  of  cloth  will 
make  10  stocks,  how  many  stocks  may  be  made  with  3j  yards  ? 


PROPORTION. 


201 


32.  A  man  traveled  ^  of  f  of  a  mUe  in  ^  of  an  hour  ;  how  far 
would  he  go  at  that  rate  in  f  of  |  of  an  hour  1  A.  lyim.  How  far 
in  f  of  24  hours?  A.  77 j  miles. 

33.  If  f  of  a  barrel  of  flour  costs  2f  times  f  of  2  dollars,  what  will 
5|  barrels  cost  ]  A.  S35.28. 

BY     RATIO^. 

34.  If  6  yards  of  cloth  cost  $8,  what  will  1  yard  cost "?  Since  3 
yards  are  f  or  ^  of  6  yards,  then  3  yards  will  cost  ^  of  the  price  of  6 
yards,  that  is  ^  of  $8,  which  is  S4.  A.  $4. 

35.  If  5  hats  cost  $41  what  will  30  cost  ?  What  part  of  5  is  30  ? 
A.  V  =6.     How  many  are  6  times  Ul  ?  A.  $246. 

36.  If  12  cows  cost  432  dollars,  what  will  8  cows  costT  What 
part  of  12  is  8  ?     How  much  is  f  of  $432 1  A.  $288. 

37.  When  112  bushels  of  wheat  cost  $168,  what  will  80  bushels 
cost  ]  What  part  of  112  is  80 1  How  much  is  -f  of  $168 1  A.  $120. 

38.  At  the  rate  of  $50  for  400  dozen  of  eggs,  what  will  1000  dozen 
cost  1  What  is  the  ratio  of  400  to  1000  ]  [See  lxii.  case  xi.]  How 
much  is  2|  times  $50.  A.  $125. 

39.  When  5  bushels  of  wheat  cost  $8,  what  will  be  the  cost  ot 
300  bushels  ]  What  is  the  ratio  of  5  to  300 1  How  many  are  60 
times  $81  A.  $480. 

40.  If  4  gallons  of  molasses  cost  9|  shillings,  how  many  dollars  will 
40  gallons  cost  1     Ratio  10.  A.  $10,041. 

41.  Suppose  a  man  travels  187^  miles  in  5  days,  how  far  will  he 
travel  in  25  days  1     Ratio  5.  A.  937|  miles. 

42.  If  1  bag  of  salt  cost  $5,  what  will  $500  purchase  1  What  part 
of  5  is  500  ?  A.  100  bags. 

43.  If  15  gallons  of  oil  cost  $26|,  what  will  4  gallons  cost  1  How 
muchisy'Vof$26|?  A.  7^V  or  ^7.033  +  . 

44.  If  2cwt.  2qr.  of  sugar  cost  $15,625,  what  will  50cwt.  cost? 
[See  LXII.  case  xii.]    What  part  of  10  qr.  is  200  qr. ,?  A.  $312.50. 

45.  Suppose  a  stage  runs  at  the  rate  of  7  miles  and  4  furlongs  in 
45  minutes ;  how  long  will  it  be  in  running  8m.  6fur.  ?  A.  52|min. 

46.  If  £1.  17s.  6d.  will  purchase  20  gallons  of  wine,  how  many 
gallons  will  18s.  9d.  purchase  ?  A.  10  gallons. 

47.  If  f  of  a  barrel  of  rice  cost  $17f,  what  will  750  barrels  cost? 

The  ratio  is  •^=(by  lxii.  case  x.)  1000.  A.  $17,600. 

48.  If  f  of  a  barrel  of  wine  costs  $30,  what  will  f  barrel  cost  ? 
What  part  of  f  is  f  ?  [See  lxii.  case  xi.  16.]  How  much  is  f  of 
$30?  A.  $26|. 

49.  If  I  of  a  yard  of  silk  cost  |  of  a  dollar,  what  will  vf  of  a  yard 
cost? A.  $14  or  $5.9895.+ 

Q.  When  6  loaves  of  bread  are  bought  for  48  cents,  why  do  3  loaves,  at  that 
rate,  cost  24  cents  ?  When  20  papers  of  pins  cost  120  cents,  what  will  10 
papers  cost,  and  why  ?  When  l  of  3.  of  a  dollar  buys  8  skeins  of  silk,  how 
many  skeins  may  be  bought  for  |  of  a  dollar? 

1  For  the  meaning  of  Ratio,  and  the  rule  for  finding  it,  see  liii.  case  xi. 


1st.    2nd. 

3d. 

Yds.  Yds. 

cts. 

8:24: 

:  6  3 

6  3 

7  2 

1  4  4 

202  ARITHMETIC. 

50.  When  3f  pounds  of  butter  cost  75  cents,  what  will  2|  pounds 
cost?  Whatpartof3|is2n  How  much  is  f  of  75 1  ^.  56}cents. 

51.  If  I  of  a  ship  cost  $20,000,  what  will  f  of  her  be  worth  at  that 
rate  ?    What  part  of  |  is  f  1  A.  $24,000. 

52.  How  many  yards  of  cloth  which  is  f  yd.  wide,  are  equal  to  5 

yards  which  is  |yd.  wide  1    Ratio-^or  ^.  ^.  5|  yards. 

53.  When  ^  of  |  of  a  pound  of  butter  costs  12|^  cents,  what  will 
40  firkins,  each  containing  25  pounds,  cost  1  A.  $333.33^. 

BY    S  T ATEMENT . 

54.  If  8  yards  of  cloth  cost  63  cents,  what  will  24  yards  cost  ? 
By  Ratio.— 24  yards  will  cost  V  of  63  cents.  =  $1.89.  A. 

By  Analysis. — 1yd.  cost  |  of  63cts=7|ct.x  24yds.  =$1.89  A.  :  or 
to  avoid  the  fraction,  multiply  by  the  24  first,  and  divide  by  the  8  after- 
wards ;  but  before  doing  this  a  statement  is  often  made  of  the  terms 
employed,  as  follows : — 

Observe  that  the  1st  and  2nd  terms  are  of 
the  same  kind,  and  the  3rd  term  of  the  same 
kind  with  the  answer ;  also  that  the  2nd  term 
is  the  multiplying  number,  and  the  1st  term 
the  dividing  number:  all  of  which  must  be 
observed  in  every  statement. 

Notice  also  the  colons  between  the  different 

8)1512  terms.     These  colons  are  in  common  use,  to 

$1.89  A.  show  that  the  ratio  of  8  to  24,  which  is  ( V  =) 

■  ■     ■  -     •  3,  is  the  same  .as  that  of  63  to  the  answer  89, 

which  is  also  (Vf=)  3. 

55.  If  419  books  cost  $1,257,  what  will  750 cost!     A.  $2,250. 

56.  If  750  books  cost  $2250,  what  will  419  cost  1      A.  $1,257. 

57.  If  750  books  cost  S2,250,  how  many  will  $1,257  buy  ?  A.  419. 

58.  You  doubtless  have  noticed  that  the  greater  the  multiplying 
term  in  comparison  with  the  dividing  temi^  the  greater  is  the  answer, 
and  the  reverse. 

59.  Hence  we  have  the  following  direction,  which  will  greatly 
assist  you  in  arranging  the  first  and  second  terms  : 

60.  Take  the  greater  of  these  terms  for  the  second  term  if  the 
ansicer  ought  to  be  greater  than  the  third  term,  otherioise  take  the 
smaller  for  the  second  term. 

61.  If  Icwt.  of  iron  costs  $8.25,  what  will  27cwt.  cost?  Icwt. 
27cwt.  :  :  $8.25.  A.  $222.75. 

62.  If  27cwt.  of  iron  cost  $225,  what  will  Icwt.  cost  ?    A.  $8.25. 

63.  If  35  cows  cost  $700,  what  will  89  cows  cost  ?     A.  $1,780. 

Q.  When  8  yards  of  cloth  cost  63  cents,  how  do  you  find  by  ratio  what  24 
yards  will  cost?  54.  How  is  the  same  question  performed  by  analysis  ?  54. 
How  is  it  done  by  statement  ?  54.  Which  terms  must  be  of  the  same  kind  ?  54. 
What  must  the  third  term  be  like  ?  54.  Which  term  is  the  multiplying  num- 
ber? 54.  Which  the  dividing  number?  54.  What  is  meant  by  the  colons 
between  the  different  numbers  ?  54.  What  simple  direction  is  given  in  respect 
to  the  arrangement  of  the  first  and  second  terms  ?  GO.  How  is  this  ascertained  f  58-. 


PROPORTION.  203 

C4.  If  91  horses  cost  $4,788,875,  what  will  75  cost? 

A.  $3,946,875. 

65.  If  1824  barrels  of  flour  will  cost  $15,750.64,  what  will  2736 
barrels  cost  ] 

1824  bar. :  2736  bar.::$15750.64.  But  since  1824  and  2736  have 
912  for  a  common  divisor,  we  may  use  in  their  stead  the  quotients  2 
and  3,  for  multiplying  the  third  term  by  912  and  dividing  the  result  by 
912  cannot  alter  that  term  :  thus, — 

2  :  3.:$15,750.64,  then  $15,750.64 x3^2  =  $23,625.96.  A. 

66.  When  then  the  first  and  second  terms  have  common  divisors, 
divide  by  the  greatest  divisor  and  substitute  the  quotients  for  those 
terms. 

67.  When  183,945  yards  of  cloth  costs  $674,465,872,  what  will 
147156  yards  cost]   (Gr.  com.  div.  36789.)     A.  $539,572.697yV 

68.  If  63  yards  of  tape  cost  45  cents  and  3  mills,  what  will  21  yards 
cost!  A.   15c.  Im. 

69.  If  415  bales  of  cotton  sell  in  London  for  £5,260. 2s.  6d.,  what 
are  2536  bales  worth  at  that  rate  1 

It  is  more  convenient  to 

£5,260. 2s^  6(L  =  ^^^?iS??oA  reduce  the  3d  term  to  pence. 

415  bales  :  2536  bales::  1262430  pence,     rru..  o^..,.^,.  „riii  ^f  L„>.c« 

A.  7714512d.-£32143.16s.  Jhe  answer  will  of  course 

be  pence. 

70.  Hence  when  the  third  term  is  a  compound  number : — Reduce 
it  first  to  the  lowest  denomination  in  it ;  then  proceed  as  before,  re- 
collecting that  the  answer  will  appear  in  the  same  denomination  to 
which  it  was  reduced,  which  may  then  be  brought  into  any  other  de- 
nomination required. 

71.  If  a  merchant  pay  in  London  jC56.11s.  3d.  for  25  yards  of 
broadcloth,  what  would  35|  yards  cost?  A.  J6:80.  13s.  lid. 

72.  Suppose  a  merchant  buys  2cwt.  3qr.  lOlb.  of  sugar  for  $35,625, 
how  many  hundred  weight  at  that  rate  may  be  bought  for  $468.75  ? 

A.  3,750ib.  =  37cwt.  2qr. 

73.  If  it  costs  $369,625  to  make  a  fence  over  a  distance  of  2  miles 
6fur.  16rd.  4yd.  1ft.,  what  length  of  fence  may  be  made  at  that  rate  for 
$1,108,875?  A.  8m.  3fur.  lOrd.  2yd 

74.  If  2hhd.  42gal.  2qt.  of  wine  cost  $193.20,  what  will  ,5hhd.  cost  ? 
Here  2hhd.  42gal.   2qt.=674qt.  :  5hhd.  =  1260qt.     then  674qt. : 

1260qt.::$193.20.  A.  $361,175. 

75.  Hence  when  the  first  or  second  term  is  of  a  different  denomina- 
tion, it  must  be  brought  to  the  same  by  Reduction. 

76.  If  4  yards  of  cloth  cost  $17.35,  what  will  101yd.  3qr.  costi 

A.  $441.34.+ 

77.  If  2cwt.  3qr.  101b.  of  hay  cost  $4.12|,  what  will  5T.  15cwt. 
2qr.  201b.  cost  ? A.  $167.46  + 

•  Q.  When  the  third  term  is  a  compound  number,  what  is  to  be  done  with  it  ? 
70.  In  what  denomination  is  the  answer  ?  70.  What  reduction  is  often  re- 
quired in  reference  to  the  other  terms  ?  75. 


204  ARITHMETIC. 

78.  Suppose  lOSyd.  2qr.  Ina.  of  cotton  cloth  cost  in  Manchester, 
(England)  i:4.l3s.  6|d.,  what  would  500yd.  2qr.  cost?  ^.jC21.11s.  3d. 

79.  A  gentleman  invested  $2,000  in  coal  at  the  rate  of  $8.50  for 
19cwt.,  how  many  tons  did  he  buy  T     A.  223T.  lOcwt.  2qr.  8f|lb. 

80.  Suppose  it  costs  $49  to  move  a  certain  building  38rd.  3yd.  ; 
how  many  miles  at  that  rate  may  the  same  building  be  moved  for 
$5,000?  A.  12m.  2fur.  13rd.  |yd.  1ft.  llffin. 

RECAPITULATION. 

81.  The  Rule  of  Three  is  so  called,  because  it  has  three  terms 
given  to  find  a  fourth  (the  answer) ;  which  shall  have  the  same  ratio 
to  the  third,  as  the  second  has  to  the  first. 

82.  The  FIRST  and  second  terms  are  always  of  the  same  kind, 
and  the  third  of  the  same  kind  with  the  fourth  or  answer. 

GENERAL    RULE. 

83.  State  the  question  by  making  the  third  term  of  the  same  kind 
with  the  ansiver;  then  consider  whether  the  answer  ought  to  he 
greater  or  less  than  the  third  term ;  if  greater,  make  the  second  term 
greater  than  the  first,  but  if  less,  make  the  second  term  less  than  the  first' 

84.  Reduce  the  first  and  second  terms  to  the  same  denomination, 
and  the  third  term  to  the  lowest  denomination  in  it,  then  multiply  the 
second  and  third  terms  together,  and  divide  their  product  by  the  first, 
the  quotient  ivill  be  the  fourth  term  or  answer,  in  the  same  denomina^ 
tion  with  the  third  term. 

contractions  of  the  rule. 

85.  Reduce  the  fractional  ratio  of  the  first  and  second  terms  to  its 
simplest  form,  then  multiply  the  third  term  by  it. 

86.  Or  divide  the  first  and  second  terms  by  their  greatest  common 
divisor,  then  substitute  the  quotients  for  the  terms  themselves,  and 
proceed  as  before. 

87.  Or  proceed  analytically  to  find  the  whole  by  first  finding  the 
value  of  unity. 

88.  If  17  yards  of  satinet  cost  813.75,  what  will  51  yards  cost  ? 
By  Statement.      17yd.  :  51yd.  ::  $12.75.      ($12.75x51^17=) 
$38.25.  A.  ^38.25. 
By  Ratio.     17  :  51=f^  =  3  :  then  $12.75x3=838.25. 

A.  838.25. 
By  Analysis.     $12.75-Hl7=75ct.  for  1  yard  ;  then  75ct.  x  51yd. 
=i38.25.  A,  $38.25. 

89.  If  51  yards  of  satinet  cost  838.25,  what  will  17  yards  cost? 
Ratio  ^  :  1  yd.=75cts.  A.  812.75. 

90.  If  838.25  will  buy  51  yards  of  satinet,  what  will  $12.75  buy? 
Ratio  i      75  cents  buys  1yd.  A.  17  yards. 

91.  If  812.75  will  buy  17  yards  of  satinet,  what  will  $38.25  buy? 
Ratio  3  :  75  cents  for  1yd. A.  51  yards. 

Q.  What  is  the  Rule  of  Three?  81.     What  similarity  is  there  between  the  . 
terms  1  82.     General  rule  1  83,  84.    What  are  the  three  methods  by  which  this 
rule  is  abbreviated?  85,  86,  87. 


PROPORTION.  205 

92.  When  108  barrels  of  flour  cost  $837,  what  will  43^  barrels 
cost?     Ratio  §:  Ibl.  =$7^  ^.$334.80. 

93.  Suppose  $600  bushels  of  wheat  cost  $1,200  ;  how  many 
hushels  may  be  bought  for  $7,200  1  A.  3,600  bushels. 

94.  If  $7,200  dollars  will  purchase  3,600  bushels  of  wheat,  what 
will  600  bushels  cost '?  A.  $1,200. 

95.  When  you  pay  $13.50  per  month  (=4  weeks)  for  board,  how 
much  will  pay  your  bill  for  22  weeks  ?  A.  $74. 25. 

96.  Suppose  you  give  30  bushels  of  rye  for  120  bushels  of  potatoes, 
how  much  rye  must  you  give  at  that  rate  for  600  bushels  of  potatoes  1 

A.   150  bushels. 

97.  If  4cwt.  Iqr.of  sug*  cost  $45.20,  what  will  21cwt.  Iqr.  cost  ? 
[See   ex.  75.]  A.  $226.     Ratio  5. 

98.  Suppose  you  pay  $120  for  60  yards  of  cloth;  what  does  it  cost 
by  the  ell  English  ?  A.  $2.50. 

99.  When  4  tuns  of  wine  cost  $322.56,  what  will  1  tierce  cost? 
A.  $13.44.  What  will  1  barrel  cost?  A.  $10.08.  What  will  1 
pint  cost  ?  A.  4:  cents. 

100.  When  a  merchant  compounds  with  his  creditors  for  40  cents  on 
a  dollar,  how  much  is  A's  part,  to  whom  he  owes  $2,500  ?  How  much 
is  B's  part,  to  whom  he  owes  $1,600  ?    A.  A's  $1,000  ;  B's  $640. 

101.  When  the  velocity  of  a  locomotive  on  a  railroad  is  35  miles 
an  hour,  how  far  does  it  move  in  30sec.?      A.  /jm.  or  93rd.  5^ft. 

102.  If  a  steamboat  cross  the  Atlantic  (3000  miles)  in  12  days, 
what  is  her  average  velocity  per  hour  ?  A.  lOy^miles. 

103.  The  surface  of  the  planet  Jupiter  contains  24,884,000,000 
square  miles.  How  many  inhabitants  would  it  accommodate,  if  1,120 
occupy  4  square  miles  ?  A.  6,967,520,000,000. 

104.  How  many  minutes  would  there  be  in  16  weeks,  provided 
there  were  2,160  in  3  days  ?  A.  80,640  minutes. 

105.  If  a  man's  family  expenses  are  $2.50  per  day,  and  his  salary 
$1,537.40,  with  perquisites  amounting  to  I  of  a  dollar  per  day,  how 
much  can  he  save  annually  ?  A.  $761.77^. 

106.  Jupiter  moves  in  its  annual  course  90,000  miles  every  3  hours. 
How  far  does  it  move  in  7  weeks  ?  A.  35,280,000  miles. 

107.  How  much  flour  will  a  family  consume  in  4  years,  if  275 
pounds  supply  them  37  days?  A.  55  barrels  71lbs.  5|foz. 

108.  A  man  pays  $25  for  a  load  of  corn  containing  30  bushels,  how 
many  loads  can  he  buy  for  $92.50 1        A.  3  loads  and  21  bushels. 

109.  The  breadth  of  the  dark  space  between  the  two  rings  of  Sat- 
urn is  2,839  miles.  How  long  would  sound  be  in  passing  through  it 
at  the  rate  of  1,142  feet  in  a  second  ?  A.  3h.  38m.  4:6-^\3. 

110.  The  planet  Uranus  is  1,705,000,000  rjiles  distant  when  nearest 
us.  How  long  would  a  cannon  ball  be  in  reaching  it,  moving  12,000 
miles  in  24  hours  ?  A.  389Y.  98^.d. 

111.  If  five  times  four  were  thirty-three, 

WTiat  would  the  fourth  of  twenty  be  ?  A.  8|. 

18 


206 


ARITHMETIC. 


112.  If  a  Steeple  150  feet  high  cast  a  shade  375  feet  in  length,  how 
long  is  that  staff'  whose  shadow  at  that  time  is  8  feet  T         A.  3\ 

113.  Divide  $240  in  the  proportion  of  3  to  2.  A.  $160. 

114.  Suppose  3cwt.  2qr.  101b.  of  sugar  cost  $52,625,  what  wili 
21cwt.  2qr.  lOlb.  costi  A.  $315.75. 

115.  If  you  receive  $89  interest  on  $1,780  for  one  year,  what  is 
the  rate  per  cent.;  that  is,  what  is  it  on  $100.  A.  5  per  cent. 

116.  If  12  men  build  a  wall  in  20  days,  how  many  men  can  do  the 
same  in  5  days  1  J..  48  men. 

117.  Suppose  a  wall  is  built  by  48  men  in  5  days  ;  what  number 
of  men  could  do  the  same,  if  they  were  allo.wed  to  be  four  times  as 
long  about  it  ?  •  ^4..   12  men. 

118.  If  4  men  dig  a  trench  in  48  days,  how  many  men  could  do  it 
in  the  sixth  part  of  that  time'?  A.  24  men. 

119.  Suppose  a  man,  by  traveling  10  hours  a  day,  performs  a 
journey  in  4  weeks,  without  desecrating  the  Sabbath  ;  how  many 
weeks  would  it  take  him  to  perform  the  same  journey,  provided  he 
travels  only  8  hours  per  day,  and  pays  no  regard  to  the  Sabbath  "? 

A.  4  weeks  and  2  days. 

120.  Suppose  a  certain  pasture,  in  which  are  20  cow^s,  is  sufficient 
to  keep  them  6  weeks,  how  many  must  be  turned  out,  that  the  same 
pasture  may  keep  the  rest  6  months  1  ^4.   15  cows. 

121.  If  a  certain  garrison  is  manned  with  1,000  men,  and  with  pro- 
visions enough  for  18  months,  how  many  must  leave  the  garrison, 
that  the  rest  may  be  able  to  hold  out  against  a  siege  of  2  years  ] 

A.  250  men. 

122.  Suppose  a  man  of  war  that  has  1,800  marines  on  board,  and 
provisions  enough  for  18  months,  should  lose  a  fourth  part  of  her  men, 
how  long  would  their  provisions  serve  the  rest  1  A.  2  years. 

123.  Suppose  that  50  yards  of  carpeting  1  ell  English  wide,  will 
carpet  a  room  ;  how  many  yards  of  carpeting  that  is  only  3  quarter? 
wide,  will  do  the  same  1  -4.  83^  yards. 

124.  If  7s.  6d.  in  New  Jersey  currency  is  equal  to  8s.  in  New 
York  currency,  what  sum  of  the  former  is  equal  to  £720  of  the  latter  7 

A.  £675. 

125.  A  mason  was  engaged  in  building  a  wall,  when  another  came 
up  and  asked  him  how  many  feet  he  had  laid  ;  he  replied  that  the  part 
he  had  finished  bore  the  same  proportion  to  1  league  which  tt  does 
to  87.    How  many  feet  had  he  laid  1  A.  SSyW^  feet. 

126.  A,  standing  on  the  bank  of  a  river,  discharges  a  cannon,  and 
B,  upon  the  opposite  bank,  counts  six  pulsations  at  his  wrist  between 
the  flash  and  the  report ;  now  if  sound  flies  1, 142  feet  per  second,  and 
the  pulse  of  a  person  in  health  beats  75  strokes  in  a  minute,  vi^hat  is 
the  breadth  of  the  river  I  A.  5,481fft.  or  Im.  201|  ft. 


RULE    OF    THREE    IN    FRACTIONS.  207 

RULE    OF    THREE    IN    FRACTIONS. 

GENERAL    RULE. 

LXXXVIII.  1.  State  the  question  and  ferform  the  operation  as 
before,  except  you  are  to  multiply  and  divide  the  terms  according  to 
the  rules  to  which  the  numbers  respectively  belong. 

2.  If  I  of  a  barrel  of  flour  costs  88.40,  what  will  U\h\.  cost? 
^U  :  13^bl.  :  :  $8.40.  For  dividing  by  I  see  lxvi.  18.       A.  $126. 

Or  1  barrel  costs  f  of  88.40  =  $9.60x  13^bl.  A.  S126. 

131 
Or  the  ratio  of  I  to  13 J  =-7—=  15 x  $8.40.  A.  -$126. 

3.  If  fV  of  a  dollar  will  buy  480  pins,  how  many  dozen  pins  will  $2 
purchase  1     Ratio  lOf.  A.  420f  dozen. 

4.  Suppose  you  pay  5s.  3d.  for  |  of  a  gallon  of  oil ;  what  will  19 
barrels  3  gallons  costl     Ratio  803  :  lgal.=7s.  A.  $701.75. 

5.  If  $5  will  purchase  30  yards  of  calico,  how  many  yards  will  f 
of  a  dollar  purchase  ]     Ratio  ^5.  A.  4  yards. 

6.  When  ^  of  a  hogshead  of  wine  is  bought  for  $8.50,  what  does  | 
of  a  pint  cost  1  Either  reduce  by  lxii.  case  xiv.  §pt.  to  the  fraction 
of  a  hogshead,  (==2/^0-— t^Vo);  o^  ^hhd.  to  the  fraction  of  a  pint; 
then  -^hhd.  :  ygVohbd.' :  $8.50.  A.  O/o^  mills. 

7.  When  £§  will  purchase  in  London  24  dozen  steel  pens,  how 
many  pens  will  f  of  a  penny  purchase  ?  ^-  If  pens. 

8.  If  f  of  a  bushel  of  wheat  cost  yf  of  a  dollar,  what  will  f  of  a 
dollar  purchase  ?     [See  lxvi.  18.]  ..4. -^f  bushel. 

9.  What  will  5^  yards  of  broadcloth  cost  in  London,  if  |  of  a  yard 
cost  £^^  ^  A.  £lif. 

10.  If  52f  yards  of  cloth  cost  $75^,  whiit  will  3,676-^^  yards  cost? 
52.4  yards  :  3676.7  yards  :  :  $75.50."  A.  $5297.535  +  . 

11.  If  37ilb.  of  sugar  cost  $5},  what  will  205flb.  cost? 

A.  $28,784. 

12.  When  40|  acres  of  land  are  purchased  for  $219^,  what  w^ill  5 
farms  cost  at  the  same  rate,  each  farm  containing  10.51  acres  ?  First 
reduce  all  the  terms  to  decimals.  A.  $5260.672. 

13.  If  61b.  3oz.  of  silver  will  make  2  silver  tankards,  how  many 
such  tankards  would  5131b.  3oz.  15dwt.  make?  Reduce  the  first 
and  second  terms  to  decimal  expressions  of  the  same  denomination. 
See  Lxix.  case  iv.  A.  164y^  tankards. 

14.  What  will  b\  yards  of  satin  cost,  if  f  of  a  )^ard  costs  7s.  6d.? 
Since  |yd.  =  . 625yd. :5iyd.  |yd.  :  5-J-yd.  :  :  90d.  A.  £3.  6s. 

=  Vyd.  =  5.5yd.,  and  7s.'6d.  |yd.  :  Vy^-  =  =  90d.  A.  £3.  6s. 

=9^.  or  i;.375 ;  the  num-  |yd.  :  Vyd.  :  :  V'd-  A.  £3.  6s. 

bers  are  susceptible  of  the  fyd.  :  5.5yd.  :  :  90d.  A.£S.C)S. 

adjoining  statements,  each  of  |yd.  :  5.5yd.  :  :  jC.375.  A.  £3.  6s. 
which  is  to  be  performed  .625yd.  :  5.5yd.  ::  .£.375.  ^.  £3.  6s. 
without  any  farther  redaction. 

LXXXVIII,  Q.  When  ques>tious  in  tlie  Rule  of  Three  contain  fractions 
how  are  they  performed?  1, 


208  ARITHMETIC. 

15.  What  will  563^  bushels  of  early  apples  cost,  when  f  of  a 
bushel  costs  60  cents?  A.  $450.60. 

16.  If  ^  of  a  yard  of  muslin  costs  j^^  of  a  dollar,  what  will  /^  of  a 
yard  cost  ?  a.  $yV =^i  cents. 

17.  When  ^  of  f  of  a  gallon  of  wine  costs  $|,  what  will  5^  gallons 
cost?  J[.  $9.16f. 

18.  If  3  yds.  of  cloth  that  is  2^  yds.  wide  will  make  a  cloak,  how 
many  yds.  |  of  a  yd.  wide  will  make  the  same  cloak?      A.  10yds. 

19.  If  12  men  do  a  piece  of  work  in  12}  days,  how  many  men  will 
do  the  same  in  6|  days  1  A.  24  men. 

20.  A  merchant  owning  f  of  a  vessel,  sells  f  of  his  share  for  $500 ; 
what  was  the  whole  vessel  worth?  A.  $1,125. 

21.  If  l^lb.  of  indigo  cost  $3.84,  what  will  49.21b.  cost  ? 

A.  $125,952. 

22.  If  $29  J  buy  59|yds.  of  cloth,  what  will  $60  buy  ?  A.  120yds. 

23.  How  many  yards  of  cloth  can  you  buy  for  $75|^  at  the  rate  of 
267f  yards  for  $37f  ?  A,  535^  yards. 

24.  If  7  times  f  of  |  of  an  estate  be  worth  $15,000,  what  is  f  of  | 
of  it  worth?  ^.$612,244  +  . 

25.  If  I  pay  29  cents  for  j\  of  a  yard  of  broadcloth,  for  what  can 
I  buy  3|^  yards  ?  ^.$15.08. 

26.  When  you  can  buy  §  of  §  of  a  barrel  of  flour  for  31  dollars, 
what  must  you  pay  for  19|  barrels?  A.  $158.4375. 

27.  When  the  price  of  cotton  cloth  is  f  of  a  shilling,  sterling  money, 
for  5  nails,  for  what  can  30  yards  be  purchased?  A.  £2.  17s.  7|d. 

28.  When  |  of  a  dollar  will  buy  |  of  a  pennyweight  of  gold,  how 
much  can  be  bought  for  $8,000.  A.  381b.  loz.  2dwt.  20fgr. 

29.  A  man  paid  |  of  a  dollar  apiece  for  24  apple  trees,  ^  of  a  dollar 
apiece  for  30  pear  trees,  f  of  a  dollar  apiece  for  15  plum  trees; 
how  many  of  an  equal  number  of  each  kind  could  he  have  bought  for 
$10.70?  A.  12  trees. 

30.  Suppose  a  merchant  who  has  contracted  for  the  transportation 
of  3cwt.  2qr.  151b.  for  $2.40  concludes  to  forward  a  greater  quantity, 
viz.  5T.  12cwt.  Iqr.  101b.;  what  charge  for  the  latter  would  be  in 
proportion  to  what  he  agreed  to  pay  for  the  former  ?    See  Ex.  13 

A.  $73.87^ 


THE  DOCTRINE^  OF  PROPORTION. 

DEDUCED^  MAINLY  FROM  THE   TWO  PRECEDING   CHAPTERS. 

LXXXIX.  1.  From  the  preceding  exercises  it  appears  that — 
Ratio  is  the  mutual  relation  of  one  quantity  to  another  of  the  same 
kind,  and  is  indicated  by  a  colon  written  between  the  quantities,  as 
5  :  20,  the  ratio  of  ivhich  is  5^=4. 

LXXXIX.  What  is  Ratio?  1. 

1  Doctrine,  [L.  docinno.]  Whatever  is  taught ;  a  principle;  learning;  knowledge; 
the  truths  of  the  gospel. 

2  Deducep,  [L.  dedMco.'\  Drawn  from ;  inferred. 


THE    DOCTRINE    OF    PROPORTION.  ^09 

2.  The  first  term  of  a  ratio  is  called  the  Antecedent,  and  the 
second,  the  Consequent,  and  both  together  form  a  Couplet,  as 
12  :  3.     Here  12  is  the  Antecedent,  and  3  the  Consequent. 

3.  Geometrical  Ratio  expresses  the  quotient  arising  from  dividing 
the  consequent  by  the  antecedent.  This  quotient  or  its  equivalent 
whole  number  is  sometimes  called  the  Index  or  Exponent  of  the  ratio, 
as  5  :  20,  whose  index  would  be  ^/— 4. 

4.  Quantities  between  which  there  exists  ratio  must  be  of  the  same 
kind,  or  else  we  cannot  form  any  judgment  of  their  equality  or  ine- 
quality. Thus,  2  hours  and  6  barrels  have  no  ratio  one  to  the  other, 
for  neither  can  be  said  to  be  either  greater  or  less  than  the  other. 

5.  What  is  the  ratio  of  5  to  405  !  A.  b  :  405,  or  ^-^=81.  • 

6.  Which  has  the  greater  ratio,  6  :  72  or  5  :  601  A.  Each  =  12. 

7.  Which  couplet  is  the  greater  one,  4  :  83  or  8  :  16G 1 

A.  Each=20f. 

8.  What  is  the  difference  between  the  ratio  of  3  yards  to  15  yards 
and  that  of  5  yards  to  75  yards  1  A.  10. 

9.  How  much  greater  is  the  couplet  4  :  3  than  the  couplet  5  :  3  *? 

A.  ^V 

10.  What  is  the  difference  between  2  couplets  which  have  each  40 
for  its  antecedent,  but  120  and  1,800  for  their  consequents.      A.  42. 

11.  What  is  the  difference  between  the  couplet  2  yards  :  1  yard 
and  the  couplet  1,800,000  yards  :  900,000  1  A.  0. 

12.  Hence  small  quantities  may  have  the  same  ratio,  and  even  a 
greater  one,  than  quantities  many  times  larger ;  from  which  it  is 
clear,  that  the  ratios  of  any  two  quantities  can  be  predicated  only  on 
their  relative  magnitude. 

13.  For  example,  the  sun  may  have  a  less  ratio  to  the  moon  than 
a  bullet  to  a  small  particle  of  matter,  though  the  former  quantities  are, 
either  of  them,  immensely  greater  than  the  latter. 

14.  When  two  couplets  have  the  same  ratio,  they  are  said  to  form 
a  proportion,  and  the  terms,  to  be  proportionals. 

15.  Hence,  Proportion  may  be  denominated  an  equality  op 
ratios,  and  is  denoted  usually  by  two  colons  pi^aced  between 
two  couplets. 

16.  For  example,  2  :  10  :  :  8  :  40,  which  is  read  thus : — 
The  ratio  of  2  to  10  is  equal  to  the  ratio  of  8  to  40. 
Or  the  ratio  of  2  to  10  is  the  same  as  that  of  8  to  40. 
Or  2  has  the  same  relation  to  10  that  8  has  to  40. 

Or  2  is  contained  in  10  as  often  as  8  is  in  40  ;  for  5  times  3 
are  10,  and  5  times  8  are  40. 

17.  The  proportional  terms  take  their  names  from  their  location ; 

Q.  What  names  have  the  terms?  2.  Give  an  example.  How  is  geometrical 
ratio  expressed?  3.  What  is  essential  to  the  existence  of  ratio?  4.  What  is 
the  ratio  of  5  to  20?— of  22  to  11  ?— of  8  to  30?— of  \  to  J  ?— of  -^^  to  f  t 
When  are  four  numbers  proportional?  14.  What  then  is  Proportion?  15. 
What  are  the  different  methods  of  reading  examples  ?  IG.  How  are  the  pro 
DOTtional  terras  distint;uished  ?  17.  Give  an  example,  [See  18.] 
18* 


210 


ARITHMETIC. 


thus,  the  first  and  last  are  called  the  extreme  terms,  or  the  Ex- 
TREMES,  and  the  second  and  third,  the  middle  terms,  the  mean  terms, 
or  the  Means. 

18.  Thus  in  2  :  6  :  4  :  12,  the  2  and  12  are  the  extremes,  and  the 
6  and  4  the  means. 

19.  Proportion  generally  consists  of  four  terms,  but  it  may  exist 
with  only  three,  when  the  first  number  has  to  the  second  the  same 
ratio  as  the  second  has  to  the  third,  which  is  thence  called  continued 
proportion  ;  as,  2  :  :  6  :  18 ;  for  3  times  2  are  6,  and  3  times  6  are  18. 

20.  Hence,  we  need  not  hesitate  to  pronounce  any  set  of  numbers 
proportional,  when  we  can  prove  that  an  equality  of  ratios  exists  be- 
tween them. 

21.  Arithmetical  Ratio  is  the  difference  between  two  quantities, 
as  6 — 3,  and  is  denoted  by  the  sign  between  the  couplets. 

22.  For  example,  7—5=8—6  is  a  proportion;  for  the  ratio  of 
each  couplet  is  2,  and  the  extremes  7  and  6  added  together  are  equal 
to  the  means  5  and  8  added  together,  and  universally — 

23.  In  Arithmetical  Proportion  the  sum  of  the  extremes  is 

EQUAL  TO  the  SUIVI  OF  THE  MEANS. 

24.  Geometrical^  Proportion  is  an  equality  of  Geometrical 
Ratios,  and  Arithmetical  Proportion  an  equality  of  Arithme- 
tical Ratios. 

25.  The  terms  Geometrical  and  Arithmetical  are  generally  used  as 
above,  because  they  are  employed  in  the  same  sense  in  Geometry^ 

26.  In  the  proportion  4:3:  :  8  :  6,  the  ratios  f  and  f  are  equal ; 
for  I  reduced  is  equal  to  f . 

27.  Again,  if  the  ratios  f  and  f  are  equal,  it  follows,  that,  by  re- 
ducing them  to  a  common  denominator,  their  numerators  will  become 
equal,  as  before.  This  is  actually  the  case,  for  they  make  f|  and  f |, 
but  the  first  numerator,  24,  is  in  reality  the  product  of  the  two  ex- 
tremes, 4  and  6,  and  the  second  numerator  the  product  of  the  two 
means,  3  and  8, 

28.  Hence  we  deduce  an  important  principle,  viz.  That  in  every 
geometrical  proportion  the  product  of  the  two  extreme  terms 
IS  equal  to  the  product  of  the  two  mean  terms. 

29.  For  example,  4  :  5  :  :  8  :  10.  Here  the  product  of  the  two 
extremes,  4  and  10,  is  40,  and  the  product  of  the  two  means,  5  and  8, 
is  also  40.     The  numbers  are  therefore  proportional. 

Q.  Of  how  many  terms  does  Proportion  consist?  19.  What  is  an  Arithme- 
tical Ratio?  21.  Give  an  example.  See  21.  What  is  said  of  the  sum  of  the 
extremes?  What  is  Geometrical  Proportion?  24.  Arithmetical  Proportion?  24. 
Wlience  the  origin  of  these  terms  ?  25  How  is  the  proportion  4  :  3  :  :  8  :  6 
proved  to  be  such?  26.  What  other  proof  is  adduced?  27.  In  the  last  process, 
which  terms  are  multiplied  together?  27.  What  important  principle  is  deduced 
from  the  operation  ?  28.     Give  an  example  ?  29. 

1  Geometrical.  According  to  the  rules  or  principles  of  geometry.' 

2  Geometry,  [Geometria.']  The  science  of  magnitude  3  in  general,  comprehending 
the  doctrine  and  relations  of  whatever  is  susceptible  of  being  increased  or  diminished. 

3  Magnitude,  ih,  magnitudo.^  Extent  of  dimensions  of  parts;  extent;  bulk;  gran 
deur;  importance. 


THE    DOCTRINE    OF    PROPORTION. 


211 


30.  From  the  principle  just  deduced  it  follows,  that  the  order  of  the 
terms  of  a  proportion  may  be  changed,  provided  they  be  so  placed  that 
the  product  of  the  extremes  shall  equal  the  product  of  the  means ;  thus : 


80  for  10X80  =  40X20. 
80for40x20  =  80x  10. 
40  for  20X40  =  10X80. 
40  for  20X40  =  80X10. 
20for40x20  =  80x  10. 
20  for  40X20  =  10X80. 
lOforSOX  10  =  40X20. 
10  for  80x10  =  20x40. 


Each  ratio  4. 
Each  ratio  2. 
Each  ratio  2. 
Each  ratio  4. 
Each  ratio  2 
Each  ratio  4. 
Each  ratio  2. 
Each  ratio  4. 


1 0 : 40  :  :  20 
10:20:  :40 
20:  10:  :80 
20 : 80  :  :  1  0 
40: 80:  :  10 
4  0  :  1  0  :  :  8  0 
80:40:  :20 
8  0  :  2  0  :  :  4  0 

31.  Hence  proportionals  in  changing  places  may  vary  their  ratios, 
but  observe  that  their  equality  is  maintained,  which  is  the  true  char- 
acteristic of  every  proportion. 

32.  Both  antecedents,  or  both  consequents,  or  even  all  the  terms 
of  a  proportion,  may  be  multiplied  or  divided  by  the  same  number 
without  disturbing  the  equality  of  the  ratios ;  consequently  the  terms 
will  still  be  proportional. 

33.  For  the  principle  of  equality  cannot  be  affected  by  the  process, 
because  in  each  instance  of  multiplying  the  same  factor  is  contained 
in  each  product,  and  in  each  instance  of  dividing,  the  same  factor  is 
cancelled  from  each  dividend,  as — 

4  :       8  :  :      6  :  1  2  the  given  proportion. 

5 5 Multiplying  antecedents  by  5  ; 

2  0:       8  :  :  3  0  :   1  2  for  20  x  12=8x30. 

4 4  Multiplying  consequents  by  4 ; 

4  8  for  20X48=32X30. 


2  0 
1  0 


3  2 


3  0 
1  0 


3  2 

8 


Dividing  antecedents  by  10; 

4  8  for  2x48=32x3. 

8  Dividing  consequents  by  8 ; 


3  6 
3 


3  :       6  for  2x6=4x3. 

9  :       9  Multiplying  all  the  terms  by  9 : 

2  7  :  5  4  for  18x54=36x27. 
3  3  Dividing  all  the  terms  by  3. 


6:12::       9:18  for  6  x  18  =  12x9. 

34.  Two  geometrical  proportions  may  be  multiplied  or  divided  one 
by  the  other,  term  by  term,  with  results  still  proportional : 

35.  For  the  fractional  ratios  of  each  couplet  being  equal,  the  pro- 
cess is  the  same  in  effect  as  multiplying  or  dividing  equal  fractions  by 
equal  fractions,  the  results  of  which  will  of  course  be  equal ; 


1st  Prop.  3 
2nd  Prop.  2 

:       6  : 

8  : 

:        5  : 
3  : 

1  0  Multiplying  the  1st 
1  2  and  2nd,  term  by  term 

3d    Prop.  6 
1st  Prop.  3 

4  8  : 
6  : 

:   1   5  : 
:        5  : 

1  2  0  for  6X120=48X15. 
1  0  Dividing  by  the  1st, 

2nd  Prop.  2 

8   : 

:        3   : 

1  2  for  2x12=8x3. 

36.  The  terms  of  one  proportion  may  be  added  to  or  subtracted 


212 


ARITHMETIC. 


from  the  corresponding  terms  of  another  proportion,  on  the  principle, 
that  if  equal  fractions  be  either  taken  from  or  added  to  other  equal 
fractions,  the  results  must  be  equal. 
1st   Prop.  5  :        7  :   :  1  0  :   1  4 

2nd  Prop.  3         4  6         8        SubtracttheSndfromthelst, 

3d    Prop    2~'       3~^^       T~-      iB    *^^^  ^^^  ^^®  three  together; 

10:14  :  i^oVTS     for  10X28=14X20. 

37.  Note. — ^We  have  seen  that  a  number  is  squared  by  multiply- 
ing it  by  itself,  [viii.  22,]  and  cubed  by  multiplying  its  square  by  the 
same  number  again,  [viii.  27.]  These  terms  have  a  general  name, 
called  powers  ;  thus  5  before  it  is  multiplied  is  called  the  first  power ; 
6  X  5  or  25,  the  square  or  2nd  power  of  5  ;  5  x  5  x  5  or  125,  the  cube 
or  3d  power  of  5 ;  and  5  x  5  x  5  x  5  or  625,  the  4th  power  of  5,  and 
so  on,  every  power  being  named  from  the  number  of  times  the  given 
number  is  used  as  a  factor. 

38.  Proportionals,  we  have  seen,  may  be  multiplied  by  themselves 
term  by  term ;  that  is,  they  may  be  squared,  or  cubed,  or  raised  to 
any  power  whatever  without  afiecting  the  proportion ;  as — 

1st  proportion,  4  :  3   :  :  8  :  6 

Squared,  16:  9  :  :  64:  36 

Cubed,  6  4:       27::  512 

4th  power,  256:       81::       4096 

5th  power,  1024:241::32768 

39.  Proportion  is  susceptible  of  other  useful  changes;  but  thosa 
already  noticed  are  sufficient  for  our  present  purpose. 


2  1  6 
12  9  6 
7  7  7  6 


APPLICATION  OF  GEOMETRICAL  PROPORTION: 

ILLUSTRATING    MORE   FULLY   THE   RULE   OF   THREE. 

XC.  1.  Since  the  product  of  the  extremes  is  equal  to  the  product 
of  the  means,  one  product  may  be  taken  for  the  other ;  consequently,* 

2.  In  dividing  either  product  by  one  extreme,  the  quotient  will  be 
the  other  extreme,  and  dividing  by  one  mean,  the  quotient  will  be  the 
other  mean. 

3.  Find  the  value  of  x,  that  is,  find  the  fourth  term  in  the  propor- 
tion 3  :  30  ::  7  :  x.     Thus,  30X7=210^3=70.  A.  x=^10. 

4.  Find  the  third  term  or  the  value  of  x  in  the  proportion  4  :  20  : : 
sc  :  25.  A.  x—5. 

5.  Find  the  second  term  or  the  value  of  x  in  the  proportion  2^  : 
a?::  8:20.  A.  a:=6j. 

Q.  What  changes  in  the  order  of  the  terms  are  admissible  ?  30.  What  va- 
riatipn  is  therein  the  ratios?  31.  Give  an  example.  What  operations  may  be 
performed  with  those  terms  ?  32.  Why  is  not  the  proportion  destroyed  ?  33. 
On  what  principle  can  one  proportion  be  multiplied  by  another?  35.  How  may 
one  proportion  be  added  to  or  subtracted  from  another  ?  36.  Why  so  ?  36. 
What  is  meant  by  a  square,  cube,  &c.?  37.  How  may  a  r*foportion  be  squarei(, 
cubed,  &;c.?  37. 

*  From  Lacroix. 


APPLICATION   OP   GEOMETRICAL   PROPORTION.  213 

6.  What  is  the  value  of  x,  or  the  first  term  in  the  proportion  x  : 
75i::42:151?  A.  x=2l. 

7.  When  one  couplet  of  a  proportion  is  6  :  24,  and  another  563| :  x, 
what  is  the  value  of  x,  or  what  is  the  numeral  consequent  for  the 
latter  couplet  ?  J. .  a?=  2,254. 

8.  When  6  is  the  consequent  and  18  the  antecedent  of  one  couplet 
in  a  proportion,  what  will  be  the  antecedent  of  the  other  couplet  if ' 
its  consequent  be  1200  ?  ^.3,000. 

9.  When  the  first  term  in  a  proportion  is  20,  the  second  term  849, 
the  third  term  6,750,  and  the  fourth  term  x,  what  is  the  value  of  x  f 

A.  a:=  286,5371. 

10.  The  operation  by  which,  when  any  three  terms  of  a  geometri- 
cal proportion  are  given,  we  find  the  fourth,  is  called  the  Rule  of 
Three,  or  the  Rule  of  Proportion. 

11.  The  Rule  of  Three,  then,  is  based  on  the  principle,  that  the 
product  of  the  extremes  is  equal  to  the  product  of  the  means :  conse- 
quently, dividing  the  product  of  the  second  and  third  terms  by  the 
first,  as  directed  by  the  rule,  is  the  same  thing  as  dividing  the  pro- 
duct of  the  means  by  one  of  the  extremes  to  find  the  other  extreme. 

12.  If  30  barrels  of  flour  cost  $240,  what  will  90  barrels  cost  ? 

13.  In  this  example  not  only  the  ratio  of  30  barrels  to  90  barrels, 
which  is  30  :  90,  is  given,  but  also  the  antecedent  of  the  next  coup- 
let, for  the  ratio  of  the  price  must  correspond  with  the  ratio  of  the 
quantities  ;  that  is,  if  one  quantity  is  3  times  greater  than  the  other, 
it  will  cost  3  times  as  much ;  if  4  times  greater,  4  times  as  much ; 
the  question  then  may  be  resolved  into  the  following  proportion — 

30  barrels  :  90  barrels  :  :  240  dollars  :  x.  Here  the  product  of  the 
means  is  21,600,  which,  being  divided  by  one  of  the  extremes,  gives 
a  quotient  of  720.  That  is,  multiply  the  second  and  third  terms  to- 
gether, and  divide  the  product  by  the  first ;  the  quotient  will  be  the 
fourth  term  or  ansiver.  A.  $720. 

14.  If  20  pounds  of  butter  cost  S5,  what  will  80  pounds  cost  1 
20  pounds  :  80  pounds  :  :  5  dollars.  Here  the  first  consequent  is  80, 
because  80  pounds  will  cost  more  than  20  pounds.  Multiply  and 
divide  as  before,  or  multiply  the  third  term  by  the  ratio  of  the  first 
and  second.  A.  $20. 

15.  If  20  men  mow  a  meadow,  consisting  of  30'aqyes,  in  a  day, 
how  many  acres  would  40  men  mow,  at  that  rate,  in  the  same  time  1 
Here  the  more  men,  the  more  acres  will  be  mowed,  which  is  called 
Direct  Proportion,  or  a  case  in  which  more  requires  more.  A.  60A. 

16.  If  40  men  mow  60  acres  in  a  day,  how  many  acres  will  20 
men  mow  in  the  same  time  1    Here  the  less  the  number  of  men,  the 

XC.  Q.  How  may  the  absent  term  in  Proportion  be  found  ?  2.  Why  so  ?  1. 
Describe  that  operation  which  has  received  the  name  of  the  Rule  of  Three.  10. 
On  what  principle  is  it  based  ?  11.  What  reason  is  given  for  multiplying  and 
dividing  in  that  rule  ?  11.  What  is  the  method  of  stating  example  12,  and  why  ?  13. 
mrWhenever  an  example  is  referred  to  in  this  manner,  it  is  expected  the 
teacher  will  read  the  example  audibly  to  the  scholar  before  he  asks  the  ques 
tion.,ai    What  is  the  statement  of  example  14,  and  why  ? 


214  ARITHMETIC. 

less  the  number  of  acres  mowed,  being  a  case  in  which  it  is  said, 
less  requires  less,  and  is  also  called  Direct  Proportion.       A.  30A. 

17.  Direct  Proportion,  then,  is  when  one  ratio  increases  as 
another  increases,  or  decreases  as  another  decreases,  and  was  for- 
merly called  the  Rule  of  Three  Direct.* 

18.  If  20  men  mow  a  meadow  in  10  days,  how  long  will  it  take  40 
men  to  do  the  same  1  Here  the  more  men,  the  less  time  will  be  re- 
quired ;  then  double  the  number  of  men  would  require  half  as  many- 
days  ;  or  if  the  number  of  men  decreases  in  any  ratio  whatever,  the 
number  of  days  will  increase  in  the  same  ratio. 

19.  The  ratio  then  of  20  men  to  40  men  in  the  last  example  is  di- 
rectly the  reverse  of  10  days  to  the  answer ;  but  if  the  20  and  40 
change  places,  that  is,  be  inverted,  the  ratios  will  become  equal,  in 
which  case  we  can  proceed  as  before  ;  thus — 40  men  :  20  men  :  :  10 
days.  A.  5  days. 

20.  The  proportion  here  then  may  be  called  Inverse,  but  the 
method  of  stating  the  question  is  the  same  as  before. 

21.  That  is,  take  of  the  first  two  terms,  the  greater  one  for  the 
second  term,  when  the  answer  requires  it,  otherwise,  take  the  smaller 
for  the  first  term. 

22.  If  40  men  mow  a  field  in  5  days,  in  what  time  will  20  men 
mow  the  same  ]  Here  the  less  men,  the  more  days  will  be  required, 
It  being  a  case  in  which  it  is  said  less  requires  more.  The  ratio  then 
of  40  men  to  20  men  decreases  as  that  of  5  days  to  the  answer  in- 
creases. 

23.  But  if  we  invert  the  first  couplet,  we  have  20  men  ;  40  men, 
which  have  the  same  ratio  that  the  third  term  has  to  the  answer,  in 
which  case  we  may  proceed  as  before. — 20  men  :  40  men  : :  5  days. 

^.10  days. 

24.  In  the  foregoing  example  the  proportion  is  also  called  Inverse, 
but  the  method  of  stating  and  operating  corresponds  exactly  with  the 
ditections  given  above. 

"25.  Inverse  Proportion,  then,  is  when  one  ratio  increases  as 
another  decreases,  or  decreases  as  another  increases,  and  was  for' 
merly  called  The  Rule  of  Three  Inverse.* 

Q.  What  is  Direct  Proportion?  17.  What  is  the  statement  of  example  23,  and 
the  reason  for  it?  What  is  the  proportion  called  ?  20.  What  is  the  method  of  pro- 
cedure? 21.  What  is  the  statement  of  example  22  ?  What  reasons  are  assigned? 
22,  23.  What  is  Inverse  Proportion  ?  25.  What  distinction  was  formerly  made 
in  Proportion  ?  *  25.  What  was  the  Rule  of  Three  Direct  ?  *  25.  What,  that  of 
Inverse?*  25.     Which  place  did  the  term  like  the  answer  occupy?  *  25. 

*  Formerly  the  following  distinctions  obtained  in  respect  to  Direct  and  Inverse  Pro- 
portion, viz. 

The  Rule  of  Three  Direct  has  three  terms  given  to  find  a  fourth,  which  shall 
have  the  same  proportion  (or  ratio)  to  the  third  term  that  the  second  has  to  the  first. 

The  Rule  of  Three  Inverse  has  three  terms  given  to  find  a  fourth,  which  shall 
have  the  same  proportion  to  the  second  as  the  first  has  to  the  third. 

The  Rule  of  Three  Direct  is  when  more  requires  more,  or  less  requires  less.  It  may 
be  known  thus :  more  requires  more  when  the  third  term  is  more  than  the  first,  and  re- 
quires the  fourth  term,  or  answer,  to  be  more  than  the  second ;  and  less  requires  less, 
when  the  third  term  is  less  than  the  first,  and  requires  the  fourth  term,  or  answer,  to  be 
less  than  the  second. 


COMPOUND   PROPORTION.  215 


COMPOUND   PROPORTION. 

XCI.  1.  If  a  man  travels  60  miles  in  5  days,  traveling  3  hours 
each  day,  how  far  will  he  travel  in  10  days,  if  he  travels  9  hours 
each  day  1 

2.  Bv  Analysis. — If  he  travels  60  miles  in  5  days,  he  travels  in 
1  day  3  of  60,  which  is  12  miles  ;  and  if  he  travels  3  hours  each  day, 
he  travels  in  1  hour  j  of  12,  which  is  4  miles.  Then,  if  he  travels  4 
miles  in  1  hour,  he  will  travel  in  9  hours,  or  1  day,  4  times  9,  which 
is  36  miles ;  and  in  10  days,  10  times  36,  which  is  360  miles,  the 
answer. 

3.  In  the  foregoing  example  the  answer  evidently  depends  on  two 
circumstances,  viz. — the  number  of  days  the  man  travels,  and  the 
number  of  hours  he  travels  each  day.  These  circumstances  we  will 
now  consider  separately,  on  the  principles  of  Simple  Proportion. 

4.  We  will  first  enquire  how  far  he  will  go  in  10  days,  provided  he 
travel  an  equal  number  of  hours  each  day ;  this  question,  then,  may 
be  expressed  as  follows  : 

5.  If  a  man  travel  60  miles  in  5  days,  how  far  will  he  travel  at 
that  rate  in  10  daysl — which  will  form  the  following  proportion — 

5  days  :  10  days  :  :  60  miles  :  A.  120  miles. 

6.  In  the  next  place  we  will  consider  the  other  circumstance,  viz. 
the  difference  in  the  number  of  hours  ;  the  question  will  then  be — 

7.  If  a  man,  by  traveling  3  hours  a  day,  travels  120  miles  in  a  cer- 
tain number  of  days,  how  far  will  he  go  in  the  same  number  of  days  if 
he  travel  9  hours  each  day] — which  gives  the  following  proportion — 

3  hours  :  9  hours  :  :  120  miles  :  A.  360  miles. 
These  two  statements  brought  together  stand  thus — 
5  days    ;  10  days    :  :    60  miles  ;  A.  120  miles. 
3  hours  :    9  hours  :  :  120  miles  :  A.  360  miles. 

XCI.  Q.  What  is  the  solution  of  the  first  question  by  analysis  ?  2.  IHFThe 
scholar  should  be  allowed,  in  cases  like  the  last,  to  copy  the  example  referred 
to,  on  his  slate,  and  have  it  before  him  while  he  is  performing  the  operation,  or 
answering  the  questions  respecting  it.cQI  On  what  does  the  answer  of  the  first 
example  depend  ?  3.  What  is  the  first  object  of  enquiry  ?  4.  What  will  be  the 
form  of  the  question  ?  5.     Statement  ?  5.  What  is  the  next  enquiry  ?  7. 

Rule  1.  State  the  question,  that  is,  place  the  numbers  so  that  the  first  and  third  terms 
may  be  of  the  same  name,  and  the  second  term  of  the  same  name  with  the  answer,  or  thing 
sought. 

2.  Bring  the  first  and  third  terms  to  the  same  denomination,  and  reduce  the  second  term 
to  the  lowest  denomination  mentioned  in  it. 

3.  Divide  the  product  of  the  second  and  third  terms  by  the  first  term ;  the  quotient  will 
be  the  answer  to  the  question,  in  the  same  denomination  with  the  second  term,  which  may 
be  brought  into  any  other  denomination  required. 

The  Rule  of  Three  Inverse  is  when  more  requires  less,  or  less  requires  more,  and  may 
be  known  thus :  more  requires  less  when  the  third  term  is  more  than  the  first,  and  re- 
quires the  fourth  term,  or  answer,  to  be  less  than  the  second  ;  and  less  requires  more 
when  the  third  term  is  less  than  the  first,  and  requires  the  fourth  term  to  be  more  than 
the  second. 

Rule.  State  and  reduce  the  terms  as  in  the  Rule  of  Three  Direct ;  then  multiply  the  first 
and  second  terms  together,  and  divide  their  product  by  the  third  term;  the  quotient  will  be 
the  answer,  in  the  same  denomination  with  the  middle  term 


216  ARITHMETIC* 

8.  In  performing  these  examples,  we  in  the  first  place  multiplied  60 
by  10,  and  divided  the  product  by  5,  making  the  120  in  the  second 
statement ;  then  we  multiplied  the  120  by  9;  and  divided  the  product 
by  3. 

9.  But  since  the  result  will  be  the  same,  we  may  as  well  multiply 
the  60  at  once,  by  the  product  of  the  two  multipliers,  9  and  10,  and 
divide  this  result,  (5,400,)  by  the  product  of  the  two  divisors,  3  and  5, 
in  which  case  the  two  statements  may  be  incorporated  into  one,  and 
performed  as  follows — 

5  days  :   1  0  days    >     miles 
3  hours    _9  hours  >    :  :  6  0 
15  9  0  1 


Then  6  0  miles 

9  0 

)  5  4  0  0 

A.  3  6  0  miles. 

10.  Or,  since  the  ratio  of  5  to  10  is  V"?  or  2,  he  will  travel,  in  10 
days,  (other  things  being  equal,)  2  times  as  far  as  in  5  days,  that  is, 
2  times  60,  or  120  miles.  And  since  the  ratio  of  3  hours  to  9  hours 
is  I,  or  3,  he  will  travel  3  times  as  far,  when  he  travels  9  hours  each 
day,  as  when  he  travels  only  3  hours  each  day ;  that  is,  3  times  120, 
or  360  miles. 

11.  The  last  process  consists  in  multiplying  the  third  term  first  by 
one  ratio,  and  that  product  by  the  other ;  but  the  effect  is  the  same, 
if  we  multiply  the  60  at  once  by  the  product  of  the  two  ratios,  2  and 
3,  or  6,  thus— 6x60=360  miles,  answer. 

12.  Hence  the  method  of  stating  is  the  same  in  principle  as  that 
of  Simple  Proportion. 

13.  If  a  man  travel  7,800  miles  in  260  days,  traveling  4  hours  each 
day,  how  many  miles  would  he  travel  in  390  days,  traveling  8  hours 
each  day  ? 

Days    2  6  0:      3  9  0  >  miles  We  write  7,800  for  the 

Hours 4  : 8  5    :  :  7  8  0  0    3d  term,  because  it  is  like 

1040     3120  the  answer.  Then  we  take 

Then  7,800  x  3, 120-^  1,040=23,400    2  terms  of  the  same  kind, 
miles,  answer.  say  260  days  and  390  days, 

and  because  he  would  go  farther  in  390  days  than  in  260  days,  we 
write  the  greater  for  the  2nd  term  and  the  smaller  for  the  1st.  Next, 
taking  the  other  two  terms,  they  being  of  the  same  kind,  and  because 
he  would  go  farther  when  he  travels  8  hours  a  day  than  when  he 
travels  only  four  hours  a  day,  we  write  the  greater  for  a  2nd  term, 
and  the  smaller  for  a  first  term.  Then  the  third  term,  multiplied  by 
the  product  of  the  second  terms,  and  the  result  divided  by  the  pro- 
duct of  the  first  terms,  gives  the  answer. 

14.  This  process  may  be  shortened,  as  in  Simple  Proportion,  by 

Q.  How  are  the  two  statements  perfonned  by  one  operation  ?  8,  9.  How  is 
the  same  done  by  Ratio?  10,  11.  On  what  principle  does  the  statement  pro- 
ceed? 12.  How  is  example  13  done  by  statement? — ^by  analysis? — by  abbre- 
nating  the  statement  ?  14. 


COMPOUND    PROPORTION.  217 

substituting  the  quotients  arising  from  dividing  any  two  terms  by 
their  greatest  common  divisor. 

Thus  260  and  390  divided  each  by  130=2  and  3. 
And  4  and  8  divided  each  by  4=1  and  2. 
2:3)  miles 

Then  1  :  2    I  :  :  7,800.     Next  7,800x6^2=23,400  miles,  Ans. 
2:6) 

15.  Again,  since  the  two  factors  2  and  6  have  a  common  divisor,  2, 
we  may  substitute  in  their  stead  their  respective  quotients,  which  are 
1  and  3,  thus— 1  :  3  ; :  7,800 ;  then  7,800  x  3  =23,400.  A.  23,400m. 

16.  The  same  .by  Analysis. — He  would  travel  in  1  day  25-0  of 
7,800  miles,  which  is  30  miles  ;  and  in  1  hour  }  of  30  miles,  which 
is  7|-  miles  ;  then  in  390  days,  390  times  7|,  or  2,925  miles  ;  and  by 
traveling  8  hours  each  day,  8  times  2,925  miles,  which  is  23,400 
miles,  the  answer. 

RECAPITULATION. 

17.  Compound  Proportion  is  when  the  relation  of  the  required 
quantity  to  the  given  quantity  depends  on  several  circumstances 
combined. 

18.  Compound  Ratio  is  that  which  results  from  multiplying  two 
or  more  simple  ratios  together. 

19.  Compound  Proportion  is  sometimes  called  the  Double  Rule 
OP  Three,  because  it  embodies  in  a  single  process  all  those  terms 
which  by  Simple  Proportion  or  the  Single  Rule  of  Three  would 
require  two  or  more  separate  statements. 

RULE. 

20.  Select  that  number  which  is  of  the  same  kind  with  the  answer 
for  the  third  term,  and  take  of  the  remaining  numbers  any  two  of  the 
same  kind,  and  arrange  them  as  in  single  proportion  ;  then  take  two 
more  of  a  kind,  and  arrange  them  in  like  manner ;  and  so  on,  till  all 
are  used :  then  multiply  the  third  term  by  the  continued  product  of 
the  second  terms,  and  divide  the  result  by  the  continued  product  of 
the  first  terms. 

21.  When  the  terms  of  any  couplet,  or  their  products,  have  a  com- 
mon divisor,  divide  by  it,  and  substitute  their  quotients  for  the  terms 
themselves ;  after  which,  multiply  and  divide  by  the  third  term  as 
above  directed, 

22.  By  Ratio.  Multiply  the  third  term  by  the  product  of  the  ratios 
of  the  other  terms,  recollecting  to  cancel  equal  terms  in  all  practicable 
cases. 

23.  If  5  men  can  build  90  rods  of  wall  in  6  days,  how  many  rods 
can  20  men  build  in  18  days  1 

Men     5:20)     rods        5X6  =  30  :  18x20  =  360  then  90x360 
Days  6  :   18  5    :  :  90         -^30  =  1080.  A.   1,080  rods. 

Q.  What  is  Compound   Proportion  ?    17.     Compound   Ratio  ?   18.     Double 
Rule  of  Three  ?  1 9.     General  Rule  ?  20. 
19 


218  ARITHMETIC. 

24.  Or  if  we  divide  5  and  20  each  by  5,  and  6  and  18  each  by  6,  we 
shall  have  the  following  statement : — 

1  •  3^  -^^90     Then  3X4X90  =  1,080.  A.  1,080  rods. 

25.  The  same  by  Ratio. — The  ratio  of  5  to  20  is  y  or  4,  and 
thatofO  to  18  is  3;  then  4x3x90  =  1,080.  A.   1,080  rods. 

26.  The  same  by  Analysis. — 1  man  wiU  build  ^  of  90  rods,  or  18 
rods,  in  6  days,  and  in  1  day  ^  of  18  rods,  or  3  rods ;  then  20  men  wiU 
build  20  times  3  rods,  or  60  rods,  in  1  day,  and  in  18  days,  18  times 
60  rods,  or  1,080  rods,  answer. 

27.  If  10  men  can  build  a  wall  360rds.  long  in  9  days,  how  many  rods 
of  wall  could  75  men  build  in  24  days  ■?   A.  7,200  rds.=22m.  4fur. 

28.  If  a  man  travel  100  miles  in  5  days,  traveling  4  hours  each  day, 
how  far  could  he  go  in  12  days,  provided  he  travels  10  hours  each  day  ? 

A.  600  miles. 

29.  If  40  men  could  cradle,  in  10  days,  800  acres  of  rye,  how 
many  acres  could  60  men  cradle  in  15  days'?  A.  1,800  acres. 

30.  If  75  men  can  build  a  wall  7,200  rods  long  in  24  days,  how 
many  rods  of  wall  could  10  men  build  in  9  days  ]  75  men  will  build 
more  wall  than  10  men,  therefore  write  10  for  the  second  term,  and 
75  for  the  first  term.  A.  360  rods. 

31.  If  a  man  travel  100  miles  in  5  days,  traveling  4  hours  each 
day,  in  how  many  days  could  he  travel  600  miles,  provided  he  travel 
10  hours  each  day  1 

32.  By  analysis,  it  appears  that  when  he  travels  10  hours  each 
day,  he  goes  50  miles  a  day  ;  then  600-^50=12.  In  stating,  take 
notice,  that  the  more  hours  he  travels  in  a  day,  the  less  days  will  be 
required ;  therefore  make  the  second  term  the  smaller  one.  A.  12d. 

33.  If  10  men  can  build  a  wall  360  rods  long  in  9  days,  how  many 
men  would  be  required  to  build  a  wall  7,200  rods  long  in  24  days  1 
Make  9  days  the  second  term,  because  the  more  men  the  less  days. 

A.  75  men. 

34.  If  a  family  of  8  persons  spend  $480  in  24  months,  how  much 
v/ould  16  persons  spend  in  8  months  ?  A.  $320. 

35.  If  a  family  of  16  persons  spend,  in  8  months,  $320,  how  many 
persons  would  spend,  in  24  months,  $480 1  A.  8  persons. 

36.  If  4  men  receive  $24  for  6  days'  work,  how  much  would  8 
men  receive  for  12  days'  work  1  A.  $96. 

37.  If  4  men  receive  $24  for  6  days  work,  how  many  men  may  be 
hired  12  days  for  $96 1  A.  8  men. 

38.  If  $2,000  will  support  a  garrison  of  150  men  3  months,  how 
long  will  $6,000  support  4  times  as  many  men  1     Ratios  3  and  |. 

A.  2l  months. 

39.  If  $100  gain  $6  interest  in  1  year,  in  what  time  will  $900  gain 
$36  interest  1  A.  8  months. 

Q.  How  is  example  23  done  by  statement  ? — by  ratio  ? — by  analysis  ? — by 
abbreviating  the  statement  ?  24.  (The  teacher  can  select  other  examples,  to 
be  performed  in  a  similar  manner.) 


COMPOUND    PROPORTION.  219 

40.  An  usurer  put  out  $150  at  interest,  and  when  it  had  been  on 
interest  8  months,  he  received  for  principal  and  interest  $160. 
What  rate  per  cent,  did  he  receive ;  that  is,  how  many  dollars  on 
$100  for  12  months  ?  A.  10  per  cent. 

41.  When  the  amount  of  $18,000  for  2y.  6m.  15d.  is  $20,745, 
what  is  the  rate  per  cent.T  A.  6  per  cent. 

42.  Suppose  you  pay  $5,712  for  transporting  12cwt.  3qr.  400 
miles,  what  must  you  pay  for  transporting  13T.  7cwt.  3qr.  over  a  dis- 
tance of  only  75  miles?  13T.  7cwt.  3qr.  =  l,071qr.  :  12cwt.  3qr.= 
61qr.     Ratios  21  and  jV  A.  22.491. 

43.  When  you  pay  $22,491  for  transporting  13T.  7cwt.  3qr.,75 
miles,  what  must  you  pay  for  transporting  12cwt.  3qr.,400  miles T 

A.  $5,712. 

44.  When  the  transportation  of  two  boxes,  each  weighing  2cwt. 
3qr.  51b.  200  miles  costs  $5.60,  what  will  be  the  cost  of  transporting 
2T.  4cwt.  3qr.  51b.  150  miles  T     Ratios  8  and  f .  A.  $33.60. 

45.  If  45  yards  of  cloth,  5  quarters  wide,  will  make  10  suits  of 
clothes,  how  many  pieces,  each  containing  25  yards,  but  only  3  quar- 
ters wide,  will  be  required  to  make  50  suits  1  A.  15  pieces. 

46.  If  25  men  can  dig  a  trench  36  feet  long,  12  feet  broad,  in  9 
days,  in  how  many  days  would  15  men  dig  a  trench  of  the  same 
depth,  but  48  feet  long,  and  only  8  feet  broad  1 

15  men     :  25  men     ^,  ^  The  less  men,  the  more  days. 

36  length  :  48  length  >  ^^^    For  7  The  more  length,  the  more  days. 
12  width  :    8  width  )  •  •  ^  (  The  less  width,  the  less  days. 

47.  The  multiplying  of  the  third  term  by  the  product  of  the  middle 
terms,  and  dividing  the  result  by  the  product  of  the  first  terms,  gives 
13|^  days. 

The  same  by  ratio. — ^The  ratio  of  15  to  25  is  | ;  of  36  to  48,  | ; 
of  12  to  8,  f.     The  operation,  then,  may  be'fexpressed  as  follows — 
|xixfxf=^2V  =  13^-.  A.  13}  days. 

48.  If  8  men  build  a  wall  80  rods  long  and  5  feet  thick  in  6  days, 
in  how  many  days  would  3  men  build  a  wall  of  the  same  breadth,  but 
120  rods  long  and  2  feet  thick"?  The  ratios  f,  I,  and  |,  (by  cancel- 
ing,)=f ,  then  f  x  6=  V= 9f .  "  A.  9|  days. 

49.  If  15  men  dig  a  trench  48  feet  long  and  8  feet  broad  in  14f 
days,  in  how  many  days  would  25  men  dig  a  trench  of  the  same 
depth,  but  36  feet  long  and  12  feet  wide  1  A.  9^f  days. 

50.  If  25  men,  by  working  10  hours  a  day,  can  dig  a  trench  36  feet 
long,  12  feet  broad,  and  5  feet  deep,  in  9  days,  how  many  hours  a  day 
must  1 5  men  work,  in  order  to  dig  a  trench  48  feet  long,  8  feet  broad, 
and  3  feet  deep,  in  12  days  ? 

15  men       :  25  men,      "1  The  less  men,  the  more  days. 

36  length    :  48  length,        ,  The  more  length,  the  more  hours. 

12  breadth  :    8  breadth,  ^  _^^'  The  less  breadth,  the  less  hours. 

5  depth     :    3  depth,  '  *     "  The  less  depth,  the  less  hours. 

12  days       :    9  days,      J  The  more  days,  the  less  hours. 


290  ARITHT.IETIC. 

61.  The  ratios  in  the  last  statement  are  |  x  f  x  |  x  §  x  |=  f ;  then 
3xl0=:6f  hours.  Or  the  product  of  the  first  terms  is  388,800, 
and  that  of  the  second  terms  259,200,  whicli  multipUed  by  10,  and 
the  result  divided  by  the  first  product,  gives  6|.  ^-61  hours. 

52.  If  15  men,  by  w^orking  Of  hours  a  day,  can  dig  a  trench  48  feet 
long,  8  feet  broad,  and  3  feet  deep,  in  12  days,  how  many  hours  a  day 
must  25  men  work,  in  order  to  dig  a  trench  36  feet  long,  12  feet  broad, 
and  3  feet  deep  in  9  days  1  A.  10  hours. 

53.  Suppose  that  50  men,  by  working  3  hours  each  day,  can  dig, 
in  45  days,  24  cellars,  which  are  each  36  feet  long,  21  feet  w4de, 
and  20  feet  deep,  how  many  would  be  required  to  dig,  in  27  days,  18 
cellars,  which  are  each  48  feet  long,  28  feet  wide,  and  15  feet  deep, 
provided  they  work  only  5  hours  each  day  1 

The  less  cellars,  the  less  men. 
The  more  length,  the  more  men. 
The  more  width,  the  more  men. 
The  less  depth,  the  less  men. 
The  less  days,  the  more  men. 
The  more  hours,  the  less  men. 

54.  In  the  last  example,  either  product  of  the  terms  is  48,988,800, 
and  the  ratios  just  cancel  each  other ;  the  third  term,  then,  is  the 
answer  as  it  stands.  A.  50  men. 

55.  If  80  men,  by  working  5  hours  in  a  day,  can  dig,  in  27  days, 

20  cellars,  which  are  45  feet  long,  28  feet  wide,  10  feet  deep,  how 
many  men  would  dig,  in  45  days,  36  cellars,  which  are  30  feet  long, 

21  feet  wide,  and  15  feet  deep,  by  working  only  3  hours  each  day? 

A.  108  men. 


24  cellars 

:  18  cellars," 

36  length 

:  48  length. 

21  width 

:  28  width, 

men 

20  depth 

:  15  depth. 

[  ::50 

27  days 

:  45  days. 

6  hours 

:    3  hours,  J 

CONJOINED    PROPORTION. 

XCII.  1.  Conjoined  Proportion,  or  Chain  Rule,  as  it  is  some- 
times called,  is  the  combination  of  several  ratios,  or  proportions, 
through  which  the  ratio  between  the  first  and  last  term  is  discovered. 

2.  This  rule  relates  principally  to  the  exchanges  between  different 
countries,  in  respect  to  specie,  weights,  and  measures,  but  is  applica- 
ble to  common  business  transactions. 

3.  To  find  how  much  of  the  quantity  mentioned  first  is  equal  to  a 
certain  portion  of  the  quantity  mentioned  last. 

RULE. 

4.  Call  the  first  terms  in  each  part  of  the  general  question  antecC' 
dents,  and  the  following  ones  consequents ;  then  place  the  antecedents 
tn  a  column  on  the  left,  and  the  consequents  in  another  column  on  the 
right,  with  the  odd  term  under  the  former  column. 

Q  How  is  example  46  stated,  and  why? — example  50,  and  why? — example 
53,  and  why?     How  is  the  last  performed  by  cancelling?  53. 

XCn.  Q.  What  is  Conjoined  Proportion?  1.  To  what  does  it  principally 
relate  ?  2.    What  is  the  first  rule  ?  4,  5.    What  does  it  find  ?  3. 


CONJOINED    PROPORTION.  221 

6.  Multiply  the  column  of  consequents  together  for  a  divisor,  and 
that  of  the  antecedents  for  a  dividend;  the  quotient  will  he  the  quan- 
tity sought.  Or,  first  reject  opposite  and  equal  terms,  or  opposite  and 
equal  factors,  by  dividing  by  the  greatest  common  divisor;  then  pro- 
ceed with  the  remaining  terms  as  first  directed. 

6.  If  lObl.  of  flour  may  be  bought  for  30bu.  of  wheat,  and  20bu. 
of  wheat  for  10  yards  of  broadcloth,  and  60  yards  of  broadcloth  for 
120gal.  of  brandy,  and  30  gallons  of  brandy  for  60  barrels  of  cider, 
how  many  barrels  of  flour  will  purchase  12  barrels  of  cider? 

1  Obi.  flour       =30bu.  wheat. 


30x10x120x60=2,160,000;  10 
X  20  X  60  X  30  X  12  =  4,320,000  -i- 
2,100,000=2.         A.  2bl.  flour. 


20bu.  wheat  =  10yd.  cloth. 

60yd.  cloth     =  120gal.  brandy. 

SOgal.  brandy=  60bl.  cider. 

12bl.  cider. 

But,  by  cancelling  equal  terms,  we  have  left  only — 
20bu.  wheat  :  120gal.brandy.|  Then  20  x  12=240^120=2. 
12bl.  cider.  |  A.  2bL  flour. 

Again,  if  we  reject  common  factors,  by  dividing  20  and  120  by  20, 
we  have  only  1,  12,  and  6.     Then  1  x  12=  12-H6=2bl.  flour. 

Again,  dividing  the  opposite  terms  12  and  6  by  6,  we  have  only  1 
and  2  and  1.  A.  2bl.  flour. 

7.  For  the  proof,  suppose  the  flour  to  be  worth  $6  a  barrel ;  then 
find,  on  that  supposition,  the  cost  of  a  single  one  of  each  quantity ; 
and  lastly,  find  whether  2bl.  of  flour  are  worth  just  12  barrels  of  cider. 
A.  Wheat  at  $2;  cloth,  S4 ;  brandy,  2;  cider,  ^1.  Then  2bl.  of 
flour  are  worth  12bl.  of  cider,  the  above  answer. 

8.  If  280  braces  at  Venice  are  worth  300  braces  at  Leghorn,  and 
7  braces  at  Leghorn  are  worth  4  yards  at  Boston,  (U.  S.)  how  many 
braces  at  Venice  are  worth  100  yards  at  Boston?  A.  163^yd. 

9.  If  121b.  at  Boston  are  equal  to  101b.  at  Amsterdam,  and  lOlb. 
at  Amsterdam  are  equal  to  121b.  at  Paris,  how  many  pounds  at  Bos- 
ton are  equal  to  500  at  Paris  ?  Cancel  equal  terms,  and  the  work  is 
done.  A.  500. 

10.  Suppose  200  bushels  of  wheat  in  Boston  (Mass.)  are  worth 
300bu.  in  New  York,  and  20  in  New  York  are  worth  40  in  Ohio, 
and  60  in  Ohio  are  worth  75  in  Michigan,  and  25  in  Michigan  are 
worth  30  in  Illinois,  how  many  bushels  in  Boston  are  worth  1,000 
bushels  in  Illinois  ?  ^.  222f. 

11.  Suppose  20  girls  in  a  factory  can  do  as  much  work  as  15  boys, 
and  60  boys  as  much  as  25  men,  how  many  girls  would  accomplish 
as  much  as  250  men  ]  A.  800  girls. 

12.  If  15s.  in  N.  England  be  the  same  value  as  20s.  in  N.  York, 
and  24s.  in  N.  York  the  same  as  22s.  6d.  in  N.  Jersey,  and  30s.  in  N. 
Jersey  the  same  as  20s.  in  Canada,  how  many  pounds  in  N.  England 
are  the  same  value  as  £240.  7s.  6d.  in  Canada?         A.  ^288.  9s. 

13.  If  15  quarts  of  milk  will  make  2  pounds  of  butter,  and  6  pounds 
of  butter  require  as  much  milk  as  30  pounds  of  cheese,  and  40  pounds 

19* 


ARITHMETIC. 

of  cheese  be  made  from  2  cows  in  3  days,  how  many  quarts  of  milk, 
at  that  rate,  would  25  cows  give  in  6  months?  A.  45,000  qts. 

14.  If  in  the  last  example  the  milk  sell  for  5  cents  a  quart,  the  but- 
ter for  25  cents  a  pound,  and  the  cheese  for  10  cents  a  pound,  which 
would  be  the  most  profitable,  for  the  given  time,  G  months,  the  selling 
of  the  milk,  the  making  of  butter,  or  the  making  of  cheese  1 

A.  The  making  of  cheese  is  more  profitable  than  the  selling  of 
milk  by  |,  or  $750;  and  more  profitable  than  the  making  of  butter  by 
l;  or  $1500. 

15.  To  find  how  much  of  the  thing  mentioned  last,  is  equal  to  a 
certain  quantity  of  the  thing  mentioned  first. 

16.  RULE.  Arrange  the  terms  as  before,  except  the  odd  term, 
which  place  under  the  column  of  consequents,  then  cancel,  multiply, 
and  divide  the  columns  as  directed  in  the  last  rule. 

17.  In  the  example  (No.  0,)  find  how  many  barrels  of  cider  will 
purchase  20  barrels  of  flour  ?  A.  120bl.  cider. 

18.  In  the  example  (No.  8,)  find  how  many  yards  at  Boston,  are 
worth  250  braces  at  Venice.  A.  153^^. 

19.  In  the  example  (No.  9,)  how  many  pounds,  at  Paris,  are  equal 
to  600  pounds  at  Boston  ]  A.  600. 

20.  In  the  example  (No.  10,)  how  many  bushels,  in  Illinois,  are 
worth  500  bushels  in  Boston  1  A.  2,250. 

21.  In  the  example  (No.  11,)  find  how  many  men  would  accom- 
plish as  much  as  480  girls.  A.  150  men. 

22.  In  the  example  (No.  12,)  how  many  pounds  in  Canada  are  equal 
to  i:250.12s.  6d.  in  New  England  1  A.  £208.17s.  Id. 

23.  In  the  example  (No.  13,)  find  how  many  cows  would  give  5| 
kilderkins  of  milk  in  one  day.  '  A.  40  cows. 


FELLOWSHIP. 

XCIII.  1.  Fellowship,  which  is  another  name  for  the  Rule  of 
Three,  is  employed  by  persons  in  partnership  in  ascertaining  their 
respective  gain  or  loss  in  trade,  when  these  are  in  proportion  to  the 
stock,  or  stock  and  time  together. 

2.  Single  or  Simple  Fellowship  is  when  the  stock  of  each  part- 
ner is  continued  in  trade  for  equal  periods  of  time  ;  each  one's  gain  or 
loss,  therefore,  is  evidently  in  proportion  to  his  stock  in  trade. 

3.  This  rule  may  be  applied  to  cases  of  bankruptcy  and  taxation,  in 
apportioning  the  part  of  each  person  interested. 

4.  RULE.  As  the  whole  stock  :  is  to  each  man's  stock ::  so  is  the 
whole  gain  or  loss  :  to  each  man's  gain  or  loss. 

Q.  What  is  the  second  rule  ?  16,     What  does  it  find  ?  15. 

XCIII.  Q.  What  is  Fellowship?  1.  Single  or  Simple  Fellowship?  2. 
What  other  cases  does  it  embrace  ?  3.  What  is  the  Rule  ?  4.  What  abbrevi 
ation  may  be  used  ?  5. 


FELLOWSHIP.  223 

5.  Note.  The  operation  may  oftentimes  be  much  abridged  by  Analy- 
sis, or  by  multiplying  the  third  term  by  the  ratio  of  the  other  two. 

6.  Three  men,  A,  B,  and  C,  traded  in  company  ;  the  first  put  in 
$200,  the  second  $400,  and  the  third  ^600.  They  gained  $300. 
What  was  each  man's  share  of  the  gain  1 

A's  stock  $200  )  $1200 :  $200::$300  :    $50  Ans.    A's  gain. 
B's  stock  S400  VS1200  :  S400::$300  :  $100  Ans.    B's  gain. 
C's  stock  $000  )  $1200 :  $G00::$300  :  $150  Ans.    C's  gain. 
The  same  by  ratio.     There  are  fi\^^,  jVVo,  Tio\,=h  h  h  then  ^ 
of  $300  is  $50;  ^  of  $300  is  $100  ;  ^^  of  $300  is  $150. 

A.  $50;  $100;  $150. 
The  same  by  analysis,    if  $1200  gain  $300,  then  $1  will  gain  ygVff 
of  $300,  which  is  $},  and  $200  will  gain  200  times  as  much,  which 
is  the  same  as  ^  of  $200 =$50;  ^  of  $400  =  $100;  |  of  $000  =  $150. 

A.  $50;  $100;  $150. 

7.  Three  merchants.  A,  B,  and  C,  gained  by  trading  in  company 
$200  ;  A's  stock  was  $150  ;  B's  stock  $250,  and  C's  $400  ;  what  was 
the  gain  on  $1,  and  what  each  man's  gain  1 

A.  $1 ;  then  A's  $37.50  ;  B's  $62.50  ;  C's  $100. 

8.  A,  B,  and  C  freight  a  ship  with  270  tons  ;  A  shipped  on  board 
96  tons,  B  72,  and  C  102.  In  a  storm  the  captain  was  obliged  to 
throw  90  tons  overboard.  What  was  the  loss  on  one  ton,  and  what 
the  loss  of  each  man  ?  A.  JT.;  A's  32T. ;  B's  24T. ;  C's  34T. 

9.  A  and  B  traded  in  company,  with  a  joint  capital  of  $600.  A 
put  in  $350.50,  and  B  $249.50.  They  gained  $120.  What  was 
that  on  $1,  and  what  portion  belonged  to  eachT 

A.  $]  ;  A's  $70.10  ;  B's  $49.90. 

10.  A  ship  valued  at  $25,200  was  lost  at  sea.  A  owned  ^  of  it ;  B 
i,  and  C  the  rest.  What  was  the  loss  of  each  man,  provided  an  in- 
surance of  $18,000  had  been  effected  on  her  ? 

.4.  $3  on  $1 ;  A's  $2,400  ;  B's  $3,600  ;  C's  $1,200. 

11.  A  detachment,  consisting  of  5  companies,  was  sent  into  a  gar- 
rison, in  which  the  duty  required  228  men  a  day ;  the  first  company 
consisted  of  162  men  ;  the  second,  153  ;  the  third,  144  ;  the  fourth, 
117  ;  and  the  fifth,  108.  How  many  men  must  each  company  fur- 
nish in  proportion  to  the  whole  number  of  men  ]  (The  proportion  for 
1  man  is  ^  ;  then,  ^  of  162  =  54,  first  company  ;  the  second,  51 ;  the 
third  ,48;  the  fourth,  39  ;  and  the  fifth,  36  men.) 

A.  54;  51;  48;  39;  36. 

12.  Two  men,  A  and  B,  traded  in  company,  with  a  joint  capital  of 
$1,000  :  they  gained  $400,  of  which  A  took  $300  and  B  the  remain- 
der.    What  was  each  person's  stock  1 

A.  $1  gain -on  $2^  stock ;  A's  $750;  B's  $250. 

13.  A  bankrupt  is  indebted  to  A  $350,  to  B  $1,000,  to  C  $1,200, 
to  D  $420,  to  E  $85,  to  F  $40,  and  to  G  $20  ;  his  whole  estate  is 
worth  no  more  than  $1,557.50.  What  will  be  each  creditor's  part  of 
the  property  ? 


221  ARITHMETIC. 

14.  Note.  In  adjusting  such  claims,  it  is  the  general  practice  to 
find  how  much  the  debtor  pays  on  $1,  first.  A.  $l-=$|;  A,  $175; 
B,  $500;C,  $GOO;D,  $210;  E,  S42.50;F,  $20;  G,  lo". 

15.  A  wealthy  merchant  at  his  death,  left  an  estate  of  30,000  to  be 
divided  among  his  children  in  such  a  manner  that  their  shares  should 
be  to  each  other  as  their  ages,  which  were  7,  10,  12,  15  and  16  yr's. 
What  was  the  share  of  each  ?  A.  $3,500  ;  $5,000 ;  $6,000 ;  $7,500 ; 
$8,000. 

16.  A  and  B  invest  equal  sums  in  trade  and  clear  $220,  of  which 
A  is  to  have  8  shares,  because  he  is  to  transact  the  business,  and  B 
only  3  shares  :  what  is  each  man's  gain,  and  what  allowance  is  made 
A  for  his  time  1  A  $60  ;  A  $100  for  his  time. 

17.  Assessment  of  Taxes.  A  Tax  is  a  rate  or  sum  of  money 
which  is  paid  for  the  support  of  government  by  the  citizens,  in  pro- 
portion to  their  property,  except  that  on  their  heads,  which  is  called 
a  poll  tax. 

18.  RULE.  Having'  tahen  an  inventory  or  valuation  of  all  the 
taxable  property  of  the  tow7i,  and  the  number  of  polls,  deduct  from  the 
whole  tax,  the  poll  tax  {assessed  equally  on  all)  then  find  how  much 
the  remainder  is  on  one  dollar  of  the  said  inventoiy  for  the  multiplier 
of  each  person'' s  inventory,  and  to  the  pr'oduct  add  his  poll  tax  ;  the 
sum  of  which  loill  be  his  whole  tax. 

19  Suppose  that  a  certain  town  which  has  6500,000  of  taxable 
property,  and  2,000  polls,  which  are  taxed  S.70  each,  is  assessed  at 
$21,400:— 

What  is  A's  tax,  whose  list  is  $1,400  and  2  polls  1*    A.  $57.40. 

What  is  B's  tax,  whose  list  is  $1,200  and  2  polls  1       A.  $49.40. 

What  is  C's  tax,  whose  list  is  $1,265  and  1  poll  ?        A.  $51.30. 

What  is  D's  tax,  whose  list  is  $2,125  and  3  polls  1      A.  $87.10. 

What  is  E's  tax,  whose  list  is  $3521  and  2  polls  1     A.  $142.24. 

What  is  F's  tax,  whose  list  is  $825^  and  3  polls  ?  ^.35. 12. 

What  is  G's  tax,  whose  list  is  $800yVV  and  2  polls  1       A.  33.41-i^. 

What  is  H's  tax,  whose  list  is  $3751  and  1  poll?  A.  15.71. 

What  is  I's  tax,  whose  list  is  $265x^0  and  2  polls?       A.  $12.01  ^%. 

*  2,000  polls  at  70  cts.  each=  $1,400  from  $21,400  leaves  $20,000 
to  be  assessed  on$500,000  ;  which  on  1  dollar  is  $/oVTroV  or  ^V  of 
$1  ==4  cts. ;  that  is  4  cents  on  a  dollar  Then  A's  list  being  $14,00 
X  4  cents=$56  which  added  to  $140,  A's  tax  on  2  polls  at  70  cents 
each,  makes  $57.40  for  A's  whole  tax,  as  above. 


COMPOUND  FELLOWSHIP. 

XCrV.  1.  Compound  FKLLowsHir  is  when  the  stock  of  each 
partner  is  employed  for  unequal  periods  of  time  :  each  one's  gain  or 

Q.  What  is  a  tax?  17.  What,  is  the  first  requisite?  18.  What  deduction  is 
first  10  be  made  ?  18.  How  is  the  poll  tax  apportioned  ?  18.  Describe  the  rest 
of  the  process  in  finding  an  individual's  tax.  18, 


COMPOUND    FELLOWSHIP.  225 

loss  therefore  is  in  proportion  to  both  his  stock  and  the  time  it  is  con- 
tinued in  trade. 

2.  Two  men  hired  a  pasture  for  S9  ;  A  put  in  2  oxen  for  six 
months,  and  B  3  oxen  for  5  months ;  what  ought  each  to  pay  for  the 
pasture  1 

3.  Two  oxen  for  6  months  is  the  same  as  (2X0=)  12  oxen  for  1 
month,  and  3  oxen  for  5  months  is  the  same  as  (3  x  5= )  15  oxen  for 
1  month :  thus, — 

2  X  6=  12  ^  27  :  12  :  S9  :  84  A's  Ans. 

3  X  5=  15  5  27  ;  15  :  $9  ;  $^5  B's  Ans. 

RULE. 

4.  Having  multiplied  each  mail's  stock  by  the  time  it  ivas  in  trade, 
then  say  as  the  sum  of  these  products  is  to  each  man's  product,  so 
is  the  whole  gain  or  loss,  to  each  man^s  gain  or  loss. 

5.  Three  merchants,  A,  B,  and  C,  enter  into  partnership  ;  A  puts 
in  160  for  4  mo.  ;  B  $50  for  10  mo.,  and  C  $80  for  12rao. ;  but  by 
misfortune  they  lose  $50  ;  how  much  loss  must  each  man  sustain  1 

A.  A^s  S7.058+  ;  B's  ^14,705+  ;  C's  S28.235  +  . 

6.  Three  butchers  hire  a  pasture  for  $48  ;  A  puts  in  80  sheep  for 
4mo.  ;  B  60  sheep  for  2mo  ,  and  C  72  sheep  for  5mo.  ;  what  share  of 
tjie  rent  must  each  man  pay  ?  ^.  A's$19.20;  B's  17.20  ;  C's  121.60. 

7.  Two  merchants  entered  into  partnership  for  lOmo.;  A  at  first  put 
in  stock  to  the  amount  of  $600,  and,  at  the  end  of  9  months,  put  in 
$100  more ;  B  put  in  at  first  $750,  and,  at  the  expiration  of  6  months, 
took  out  $250 ;  with  this  stock  they  gained  $386  :  what  was  each 
man's  part]  A.  A's,  $200,797  ;  B's,  $185202. 

8.  On  the  first  of  January,  A  began  to  trade  with  $700,  and,  on  the 
first  of  February  following,  he  took  in  B  with  $540  ;  on  the  first  of 
June  following,  they  took  in  C  with  $800 ;  at  the  end  of  the  year, 
they  found  they  had  gained  $872 ;  what  was  each  man's  share  of  the 
gain? A.  A's,  $384,929  ;  B's,  $250.71;  C's,  $236.36. 

XCIV.  Q.  What  is  Compound  Fellowship?  1.  In  example  2,  what  nuns 
bar  of  oxen  for  1  month  is  equal  to  the  given  number  for  the  given  time  ?  What 
is  the  Rule?  4. 


APPENDIX. 

PART   THIRD. 


PRACTICE. 

XCV.  1.  Practice  is  a  concise  method  of  answering  questions 
in  the  Rule  of  Three,  when  the  first  term  happens  to  be  unity. 

2.  Operations  in  Practice  are  conducted  principally  by  supposing  a 
price,  and  taking  aliquot  or  even  parts  of  the  same  for  the  true  price. 

[XLII.     1.] 

3.  What  will  50  bushels  of  rye  cost  at  5s.  a  bushel  1  Suppose  the 
price    were    £l    per    bushel,    then    the    50    bushels    would  cost 

5s.=£^)£50  £50 ;  but  at  5s.  per  bushel  only  5-  as  much 

■£12.10s.  foT5s.=£l 

4.  What  will  be  the  cost  of  8640  yards  of  cloth  at  the  following  prices  : 

At  10  shillings  per  yard  1  =£i  A.  £4,320. 

At  6s.  8  pence  per  yard  ]  =£-^-,  A.  jb'2,880. 

At  4  shillings  per  yard^^j^l.  A.  £1,728. 

At  3s.  4  pence  per  yard  ?  =£ j.  A.  £1,440. 

At  2s.  6  pence  per  yard1=£|.  A.  £1,080. 

At  Is.  8  pence  per  yard  ^^jG-jV-  A.  £720. 

At  Is.  3  pence  per  yard'?=£Jg-  A.  £540. 

At  1  shilling  per  yard  ]  =£35-  A.  £432. 

At  10  pence  per  yard  ?  =£-if.  A.  £360. 

At  8  pence  per  yard  1=£s\.  A.  £288. 

At  5  pence  per  yard  }  =  £-^\.  A.  £180. 

At  2|  pence  per  yard  1  =£^\ .  A.  £  90. 

6.  What  is  the  cost  of  the  following  quantities  at  the  prices  annexed  ■? 
3,150  gallons  of  oil  at  2s.  6d.  per  gallon  1  A.  £393.  15s. 

4,235  yards  of  cloth  at  3s.  4d.  per  yard  ]  A.  £705.  16s.  8d. 

2,434  bushels  of  oats  at  Is.  8d.  per  bushel  1        A.  £202. 16s..  8d. 
2,678  dozen  of  oranges  at  5  pence  for  each?  A.  £669.  10s. 

4,595  quarts  of  strawberries  at  3d.  per  quart]      A.  £57.  8s.  9d. 
7.  When  the  price  is  not  an  aliquot  part,  we  may  take  the  one 
nearest  to  it  first,  then  take  an  aliquot  part  of  that  part,  and  so  on. 

XCV.  Q.  What  is  Practice?  How  is  it  performed?  How  is  example  3 
performed  ?  3.  What  are  the  divisor  and  dividend  when  the  quantity  is  8640 
and  the  price  5s.  ?— is  10s.?— 6s.  8d.— 4s.  ?— 3s.  4d.  ?— 2s.  6d.  ?— Is.  3d.  ?— 5d.? 
When  the  price  is  not  an  ahquot  part  of  the  given  quantity,  what  is  the  direc- 
tion? 7. 


PRACTICE.  227 

8.  What  will  51  barrels  of  cider  cost  at  7s.  6d.  per  barrell     (23. 
6d.=jeior|of5s.) 

i,  i  )  £  5  1    at    £  1    per  bl.          Or  |  )  £  5  1 

jC13.15s.  at5s.  |)£12.15s 

£        6  .      7  s.  6  d .  at  2  s.  6  d  ■  £  6  .     7  s.  6  d. 

A.     jC19.      2s.  6d.  at  7s.  6d.  A.  £  1  9  .     2s.  6 d. 


9.  What  will  the  following  articles  cost  ^at  the  prices  annexed  1 — 
724  desks  at  12s  6d.  each?  A.  £452  .  10s. 
140  chairs  at  15s.  3d.  each]                            A.  £106  .  15s. 
936  razors  at  8s.  6d.  each?                            A.  £397  .  16s. 
812  books  at  3s.  9d.  each?                            A.  £152  .  5s. 

715 bonnets  at  17s.  6d.  each?  A.  £625  .  12s.  6d. 

10.  What  will  llcwt.  3qr.  131b.  of  rice  cost  at  $9.60  per  cvvt.  ? 

2  qr.=^    S  9  .  6  0=cost  of  1  cwt. 

IJl 

10  5.60     =cost  of  11  cwt. 
1  qr.=^)  4.80     =cost  of  2  qr. 

10  lb.  =f )  2.40     =cost  of  1  qr. 

21b.  =\)  .9  6     =costof  101b. 

lib.  =^)  .19  2=costof2lb. 

.  0  9  6=cost  of  1  lb. 
Ans.  $  1  1  4.0  4  8=cost  of  llcwt.  3qr.  131b. 

11.  What  would  be  the  cost  of  the  following  quantities  ? — 
25  yards  2  quarters  at  $2.40  per  yard  ?  A.  $61.20 
18  bushels  3  pecks  at  $3.60  per  bushel  ?  A.  $67.50. 

5cwt.  3qr.  51b.  at  $4.20  per  cwt.  ?  A.  $24.36. 

3cwt.  Iqr.  241b.  at  $3.60  per  cwt.  1  A.  $12,504. 

4T.  15cwt.  3qr.  at  $10.50  per  ton  ?  A.  $50.268f. 

5  e.  E.  2qr.  3na.  at  $2.75  per  ell  1  A.  $15.262yV 

12.  Suppose  a  merchant  buys  7hhd.  7gal.  2qt.  of  molasses  at 
$10.62^  per  hhd.,  and  sells  |  of  it  for  Ul^  per  hhd. ;  f  of  it  for  $12 
per  hhd.  and  the  balance  for  $15  per  hhd. ;  how  much  profit  does  he 
make  on  the  whole  1  A.  $13,457.  + 


DUODECIMALS. 

XCVI.  1.  Duodecimals  are  so  called  from  duodecim,  the  Latin 
for  twelve,  because  they  decrease  by  tivelves  from  the  left  hand 
towards  the  right. 

2.  In  Duodecimals,  the  foot  is  divided  first  into  twelve  equal  parts, 

Q.  How  is  example  8  performed?  When  the  quantity  is  llcwt.  3qr.  131b., 
what  are  the  several  divisors  ?     [See  10.] 

XCVI.  Q.  What  are  Duodecimals  ?  1.  What  is  the  integer  and  its  divi 
sions  ?  2. 


228  ARITHMETIC. 

called  inches  or  primes;  each  prime  into  12  equal  parts,  called  sec- 
onds ;  each  second  into  12  equal  parts,  called  thirds,  and  so  on. 

3.  That  is,  1  inch  or  prime  is  ^^  of  a  foot. 

1  second  is  -^  of  ■^^,  that  is  y^y  of  a  foot. 

1  third  is  ^  of  ^  of -j^^ttV^  of  a  foot. 

1  fourth  is  t\j  of  yV  of  y'a  of  r2  =  2  0?rs-  of  a  foot. 

4.  These  fractions  are  distinguished  usually  by  marks  called  ac- 
cents ;  thus  8^=  8  inches  or  primes ;  8^/=yf  4  or  8  seconds  ;  W^^=^^ 
or  8  thirds,  &c.,  each  additional  mark  denoting  an  inferior  denomina- 
tion. 

5.  Since  feet  stand  in  the  place  of  units,  feet  multiplied  by  feet 
must  give  feet ;  feet  multiplied  by  12ths  must  give  12ths,  that  is, 
incnes  or  primes,  and  so  on  as  follows  : — 

6.  Feet  multiplied  by  feet  give  feet. 
Feet  multiplied  by  primes  give  primes. 
Feet  multiplied  by  seconds  give  seconds. 
Primes  multiplied  by  primes  give  seconds. 
Primes  multiplied  by  seconds  give  thirds. 
Seconds  multiplied  by  seconds  give  fourths. 
Seconds  multiplied  by  thirds  give  fifths. 
Thirds  multiplied  by  thirds  give  sixths,  &c. 

7.  That  is,  the  product  will  always  be  of  that  denomination  which 
is  indicated  by  the  sum  of  the  accents  ;  thus,  l"/xb^///=  35^^^^^^^  or  35 
sevenths. 

TABLE    OF    SOME    OF    THE    HIGHER    DENOMINATIONS. 

12^''///''  (sixths)  ....         make  ^/^//(fifth.) 

12/////  (fifths)  ....         make  V'^'  (fourth.) 

12^///   (fourths)         ....         make  V^^  (third.) 
12^''/     (thirds)  ....         make  V^   (second.) 

12^^      (seconds)        ....         make  V    (prime.) 
12^       (primes)  ....         make  1ft.  (foot.) 

8.  The  operations  of  addition,  subtraction,  multiplication,  division 
and  reduction  of  duodecimals  are  the  same  as  of  other  compound 
numbers,  12  of  an  inferior  denomination  invariably  making  one  of  the 
next  higher  denomination,  as  in  the  foregoing  Table. 

9.  How  many  feet  are  there  in  1,685^?  A.  140ft.  5^. 

10.  How  many  primes  are  therein  140ft.  5'"^  A.  1685^ 

11.  How  many  feet  are  there  in  31,049//]  A.  215ft.  7^  5^^. 

12.  How  many  thirds  in  23ft.  4/7/^8///  ?  A.  40,4 12^/'. 

13.  How  many  feet  in  2,985,984^/////?  A.  1  foot. 

14.  How  many  feet  in  1,504,935,936^//''////'?  A.  3|  feet. 

15.  How  many  ninths  in  7  feet  ]  A.  36,118,462,464/////////. 

Q.  What  parts  of  a  foot  are  these  sub-divisions  ?  3.  How  are  these  denomi- 
nations distinguished  ?  4.  What  do  feet,  primes,  &c.,  multiplied  by  each  other, 
form  ?  6.  How  can  the  denomination  of  the  product  be  determined  1  Repeat 
the  Table. 


DUODECIMALS. 


229 


16.  Add  together  425ft.  4' 8'^'^' 5'^'^ IV'''' 9'''''' :  125ft.  3^ 4^/9^^' 
2////  3/////  7//////  and  43ft.  2'  5"  W"  3""  6'""  10""" . 

A.  593ft.  10^  1"  3'"  W""  W""  2""". 

17.  If  a  stick  of  timber  which  contains  39ft.  2'  3"  9'"  be  divided 
into  two  parts,  one  of  which  shall  contain  23ft.  8^  I"  \0"',  what  will 
the  other  part  contain  ?  A.   15ft.  6^  \"  1 V". 

18.  Suppose  that  a  person  agreed  to  furnish  at  a  certain  price,  15 
sticks  of  round  timber,  each  to  contain  in  solid  measure  ^Oft.  3^  5"Q"'; 
also  30  other  sticks,  each  measuring  101ft.  2'  6"  9'";  but  on  its  de- 
livery f  of  the  whole  was  rejected  on  the  ground  that  it  did  not  answer 
the  description  in  the  contract ;  what  was  the  quantity  received] 

A.  52T.  34ft.  5'  3". 

CROSS    MULTIPLICATION    OF  DUODECIMALS. 

19.  Duodecimals  are  principally  used  by  workmen  and  artificers  in 
ascertaining  the  square  or  solid  contents  of  their  work. 

20.  The  square  content,  we  have  seen,  [vii.  46.]  is  the  product 
of  the  length  by  the  breadth ;  and  the  solid  content,  the  square  con- 
tents, multiplied  by  the  depth  or  thickness,     [vii.  60.] 

22.  The  principle  illustrated  in  (6.)  which  see,  forms  the  basis  of 
the  following  rule. 

RULE. 

22.  Having  written  feet  under  feet,  f  rimes  under  f  rimes,  SfC^muU 
tiply  by  each  denomination  separately,  beginning  with  the  highest  of 
the  multiplier  and  the  lowest  of  the  multiplicand. 

23.  Place  those  products  that  are  of  the  same  denomination  under 
each  other,  which  will  carry  the  first  denomination  in  each  successive 
product  after  the  first,  one  place  farther  toward  the  right  than  the 
former  ;  then  the  sum  of  these  partial  products  ivill  form  the  required 
product. 

24.  In  a  stick  of  timber  20ft.  9'  long,  2ft.  b'  wide  and  2ft.  3'  thick, 
how  many  solid  feet  does  it  contain  ! 

*  For  the  value  of  each  product  see  8 ; 
recollecting  always  to  carry  by  12  ;  thus, 
2ft.  X 9^=  18^-^  12=  1ft.  6  primes;  then 
20ft.  X2ft.=  40ft.  +  1ft.  (to  carry)=41ft. 
Next  b'  X  9'=  45"=:  3'  and  9" ;  20ft  X  5^= 
100^+3/ (to  carry)  103^=  8ft.  7'  and  add 
the  products  together.  To  multiply  by 
2ft.  3^  say  2ft.  x  9^^=18^''=  I'  6"  :  2ft.  x  V 
=  2^+1^=3^  :  2ft  X50ft.=  100ft.  :  3^x9^' 
=  27///=  2^/ 3^^^  :  3^x  r=3"-^2"=5  :  3'X 
50ft.  =  150^=  12ft.  6^'. 

We  begin  on  the  left  of  the  multiplier  instead  of  the  right, 
because  it  is  more  convenient,  as  may  be  seen  by  comparing 
the  adjacent  operation  with  the  one  above,  with  which  it  cor- 
responds, except  that  we  begin  to  multiply  as  usual.t 
t  Lacroix's  method  of  illustration. 

20 


2  Oft 
2 

4  1  . 

8  . 

.     9^ 

.     5' 
6' 
T  .  9" 

6  0  . 

2  . 

1'  .  9" 
3' 

10  0. 
1  2  . 

3'      6" 

6'  .  5"  . 

3'" 

A.l  1  2  . 

9'  I  1"   . 

3'" 

2  0ft. 
2     . 

9/ 
5/ 

8    . 
4  1    . 

7/  . 
6/ 

9// 

5  0    . 

1/  . 

,  9// 

230 


ARITHMETIC. 


25.  How  many  square  ft.  in  a  board  lOfl.  8^  long  and  1ft.  5'  broad  1 

A.  15ft.  1^  4''. 

26.  In  a  load  of  wood  8ft.  4^  long,  2ft.  6^  high,  and  3ft.  3'  wide, 
how  many  solid  feet  1  A.  67ft.  8^  6^^ 

Note. — Artificers  compute  their  work  by  different  measures 
Glazing  and  masons'  flat  work  are  computed  by  the  square  foot; 
painting,  paving,  plastering,  &c.  by  the  square  yard ;  flooring,  roof- 
ing, tihng,  &c.  by  the  square  of  100  feet;  brick  work  by  the  rod  of 
16^  feet,  whose  square  is  272| ;  the  contents  of  bales,  cases,  &c.  by 
the  ton  of  40  cubic  feet ;  and  the  tonnage  of  ships  by  the  ton  of  95ft 

27.  What  will  be  the  expense  of  plastering  the  walls  of  a  room  8ft. 
6^  high,  and  each  side  16ft.  3^  long,  at  62^  cents  per  square  yard  ? 

A.  $38,368. 

28.  How  many  cubic  feet  in  a  block  4ft.  y  wide,  4ft.  6^  long,  and 
3ft.  thick  1  A.  57h.  4' 6'^. 

29.  How  much  will  a  marble  slab  cost,  that  is  7ft.  4^  long  and  Ifl. 
3^  wide,  at  $1.25  per  foot?  A.  $11,458. 

30.  How  many  cubic  feet  of  wood  in  a  load  6ft.  7^  long,  3ft.  y 
high,  and  3ft.  8^  wide  ?  A.  82ft.  5^  8^^  4^^^. 

31.  What  will  the  paving  of  a  court-yard,  which  is  70ft.  long  and 
56ft.  4^  wide,  come  to,  at  8.20  per  square  yard  1  A.  $87.63,  nearly. 

32.  How  many  solid  feet  are  there  in  a  stick  of  timber  70ft.  long, 
15^  thick,  and  18^  wide?  A.  131ft.  3'. 

33.  A  man  built  a  house  consisting  of  3  stories  ;  in  the  upper  story^ 
there  were  10  windows,  each  containing  12  panes  of  glass,  each  pane 
14^  long,  12^  wide  :  the  first  and  second  stories  contained  28  windows, 
each  15  panes,  and  each  pane  16^  long,  12''  wide  :  how  many  square 
feet  of  glass  were  there  in  the  whole  house  1  A.  700  sq.  ft. 


INVOLUTION.* 

XCVn.  1.  Involution  is  the  process  of  finding  powers.  Powers 
are  the  several  products  arising  from  multiplying  any  number  by  itself, 
and  that  product  by  the  same  number  again,  and  so  on. 

2.  Any  number  is  called  a  first  power  of  itself;  but  when  it  be- 
comes repeatedly  a  factor  in  producing  other  powers,  it  is  called 
their  root,t  because  they  seem,  as  it  were,  to  grow  out  of  it. 

Q.  By  whom  are  Duodecimals  used,  and  for  what  purpose  ?  19.  How  are  the 
square  and  solid  contents  of  any  thing  ascertained  ?  20.  What  is  the  Rule  ?  22, 23. 

XCVII.  Q.  What  is  Involution?  1.  What  are  meant  by  powers?  1.  What, 
by  first  powers,  second  powers,  &c.?  2. 

*  Involution,  from  the  Latin  in,  for  in,  and  volvo,  to  roll,  signifies  the  act  of  enrolling, 
enwrapping,  or  involving;  the  state  of  being  mixed  or  complicated;  the  raismg  of 
powers,  because  a  given  number  thereby  becomes,  by  repeated  multiplications,  involved 
in  other  numbers. 

t  Root.  The  part  of  a  plant  in  the  ground.  Figuratively,  the  bottom  or  lower  part; 
the  origin,  cause,  ancestry ;  a  primitive  word  or  theme.  The  first  power,  because  it 
forms  the  basis  of  all  the  succeeding  powers. 


INVOLUTION. 


231 


3.  The  first  product,  because  the  same  factor  is  used  twice,  is 
called  the  second  power,  or  square ;  the  next  product,  because  the 
same  factor  is  used  three  times,  is  called  the  third  power,  or  cube ; 
and  so  on,  as  follows — 

Thus  3=     3,  1st  power,  or  root. 

3x3=     9,  2d  power,  or  square. 
3x3x3=   27,  3d  power,  or  cube. 
3x3x3x3=   81,  4th  power,  or  biquadrate.* 
3x3x3x3x3=  243,  5th  power. 

4.  The  Index  or  Exponent  of  a  power  denotes  the  number  of 
times  the  root  must  be  used  as  a  factor  to  produce  that  power ;  con- 
sequently, index  is  only  another  name  for  the  number  of  the  pow^er 

5.  Powers  are  frequently  expressed  by  writing  their  indices  in 
smaller  figures  on  the  right  of  their  respective  roots ;  as — 

The  8th  power  of2  is  2^=2X2X2X2X2X2X2X2=  256. 
The  2d  power  of  5  is  b^=  5x5=  25. 
The  3d  power  of  4  is  4^=  4  x  4  x  4=  64. 

RULE. 

6.  Involve  the  given  number  or  root^  that  is,  multiply  it  by  itself , 
and  the  product  by  the  root  again,  and  so  on  till  the  root  has  been 
used  as  a  factor  as  many  times  as  are  indicated  by  the  given  power  or 
its  exponent. 

1.  What  is  the  second  power  or  square  of  13  ?  A.  169. 

8.  What  is  the  third  power  or  cube  of  18  ?  A.  5,832. 

9.  What  is  the  fourth  power  or  biquadrate  of  11  ?    A.  14,641. 

10.  What  is  the  fifth  power  of  7  ?— of  9  ?     A.  16,807 ;  59,049. 

11.  What  is  the  sixth  power  of  5 1— of  4  ?      A.  15,625  ;  4,096. 

12.  What  number  is  meant  by  3=^  ?— by  5^  ?— by  20^  1— by  7*  ?— 
by  6M  ^.  9  ;  125  ;  3,-200,000  ;  2,401 ;  7,776. 

13.  What  is  the  numerical  difference  between  2"^  and  8^  1 — ^between 
9»  and  4^  ]— between  10^°  and  20^  1   A.  448  ;  6,487  ;  9,996,800,000. 

14.  What  is  the  2d  power  of  f  1— of  .75  ]— of  f  %  What  is  the  3d 
power  of  f?— of  4.22 T  A.  ^\  .5625;  -i^',  jf^;  75.151448. 

15.  What  is  the  square  of  5^'?  A.  30|. 

16.  What  is  the  square  of  16| ?  A.  212\. 

17.  What  is  the  difference  between  the  cube  of  J-  and  the  biquad- 
rate off  off?  4  3  -4. /^V 

18.  What  is  the  numerical  value  off  1— of5i  ?  ^.  AV;  l^ef. 

19.  What  number  is  equal  to  32x4^1— to  3^x2*1  ^.576;  3,888. 

20.  What  is  the  difference  between  4*  and  4^  x  4^  1  A.  0. 

Q.  What  is  meant  by  Index  or  Exponent?  4.  Give  an  example.  What  is  the 
rule  ?  6.  What  are  the  second,  third  and  fourth  powers  sometimes  called  ?  3. 
What  is  the  square  of  20? — cube  of  3? — biquadrate  of  3? — fifth  power  of  2? — 
seventh  power  of  2? 

*  Biquadrate,  from  two  Latin  words,  bis,  twice,  and  quadra,  a  square,  is  so  called 
because  that  number  which  is  used  twice  as  a  factor  in  producing  a  square  is  used  twice 
more  in  producing  the  biquadrate  or  fourth  power. 


338  ARITHMETIC. 

21.  In  the  last  example  the  exponents  of  4'  and  4'  added  together 
make  5,  the  exponent  of  4* ;  therefore — 

22.  The  powers  of  the  same  root  are  multiplied  by  adding  thetr 
exponents. 

23.  215^X215*X215«  are  equal  to  whati  A.  215". 

24.  Involve  2* ;  that  is,  raise  it  to  the  power  denoted  by  its  expo- 
nent. A.  16.     Involve  2^  A.  64.     Involve  2^  A.  4. 

25.  What  is  the  quotient  of  2"  (=64)  divided  by  2=^  (=4)1 

A.  16=2*. 

26.  Hence  powers  of  the  same  root  may  he  divided  by  subtracting 
the  exponent  of  the  divisor  from  the  exponent  of  the  dividend. 

27.  Divide  315^'  by  315'S  and  S2^'  by  82^«.  A.  315* ;  82". 

28.  What  is  the  square  of  the  9  digits  1 

A.  15,241,578,750,190,521. 

29.  What  is  the  sum  of  the  squares  of  all  the  composite  numbers 
betvi^een  1  and  20^  A.  1,442. 

30.  What  is  the  sum  of  the  cubes  of  all  the  prime  numbers  between 
1  and  20.  A.  15,803. 

31.  Suppose  there  is  a  pile  of  wood,  whose  dimensions,  that  is,  its 
length,  breadth,  and  depth,  are  each  17  feet ;  how  many  cords  does 
the  pile  contain]  A.  88c.  49ft. 

32.  Suppose  a  piece  of  land  lies  in  the  form  of  a  square,  and  each 
side  measures  135  rods  ;  how  many  acres  does  it  contain  ? 

A.  113A.  145rd. 

33.  Suppose  a  pile  of  wood,  whose  dimensions  are  each  18  feet, 
be  sold  for  $8|  per  cord  ;  what  will  the  pile  bring  1 

A.  $398,672,  nearly. 

34.  If  the  amount  of  $1  at  compound  interest  for  1  year  is  $1.00, 
what  is  the  amount  for  4  years  1 — for  5  years  ? — for  7  years  1 — for  10 
years'?    A.  $1.262477  +  ;  $1.338225  +  ;  $1.50363  +  ;  $1.790848  +  . 

35.  What  is  the  difference  in  value,  at  $18^  per  acre,  between  a 
quantity  of  land  containing  250  square  miles  and  one  which  is  250 
miles  square?  [See  vii.  44.]  A.  $747,000,000. 

36.  If  a  solid  block  of  granite,  27  feet  long,  13|  feet  wide,  and  13| 
feet  thick,  be  halved,  what  will  be  the  value  of  .19  of  each  part,  at 
the  rate  of  18f  cents  for  3  solid  feet?  A.  $29.2169  +  . 

Q.  What  is  the  numerical  difference  between  5  whose  index  shall  be  3,  and 
12  whose  index  shall  be  2  ?  What  is  the  product  of  15  whose  index  shall  be  8, 
if  muhiplied  by  23  whose  index  shall  be  5?  What  is  the  rule  for  it?  22.  What 
is  the  quotient  of  4«  divided  by  4^  ?  What  is  the  rule  ?  26.  How  many  square 
rods  in  a  plat  of  ground  12  rods  square  ?  What  is  the  difference  in  square  yards, 
between  11  square  yards  and  11  yards  square?  [See  vii.  43,  44.] 


EVOLUTION. 


233 


EVOLUTION.* 

XCVIII.  1.  Evolution  is  the  finding  of  the  root  from  having 
the  power  given,  and  is  therefore  the  converse  of  Involution,  which 
is  the  finding  of  the  power  from  having  the  root  given. 

2.  Thus,  the  sectind  or  square  root  of  36  is  6,  because  the  second 
power  or  square  of  6  is  36  ;  the  third  or  cube  root  of  27  is  3,  because 
the  third  power  or  cube  of  3  is  27  ;  the  fourth  root  of  16  is  2,  because 
the  fourth  power  of  2  is  16,  &c. 

3.  A  Root,  then,  of  any  number  is  that  factor  which,  multiplied 
into  itself  a  certain  number  of  times,  will  produce  the  given  number. 
The  process  of  finding  it  is  called  its  Extraction. 

4.  The  number  or  name  of  the  root  corresponds  with  the  number 
or  name  of  its  power. 

5.  That  is,  if  4  be  the  second  power  or  square  of  2,  then  2  is  the 
second  or  square  root  of  4  ;  and  if  27  be  the  cube  of  3,  then  3  is  the 
cube  root  of  27. 


of 


6.  Find  by  trial  the  square  root  of  64 1— of  144  ?— of  3,600 1 
.25 1— of  42.25 1— of  j%  ?— of  }  of  i  1— of  6^  1— of  3yV  ? 

A.  8;  12;  60;  .5;  6.5;  J;  i;  (6i  =  V)ior2i;   If. 

7.  Find  by  trial  the  cube  root  of  1  ]— of  8 1— of  27]— of  64 1— of 
.125  ?-of  i,  ?— of  tAo  ^— of  31 1 

A.   1;  2;  3;  4;  .5;  f;  /^  (-1);  1|. 

8.  Find  by  trial  the  biquadrate  or  fourth  root  of  16 1— of  10,000 1 — 
of  If  ?— of  yV^— of  .0016^— of  yV  of  ToVo  ^ 

A         9.      -1^.2.1.        O.         1 

9.  In  Involution,  the  required  power  of  any  number  may  be  exactly 

XCVIII.  Q.  What  is  Evolution?  1.  Give  an  example.  What  is  the  root 
of  any  number?  3.  Whence  their  names?  4.  Give  an  example.  Give  the 
answers  to  the  examples  (on  being  read  aloud  by  the  teacher)  in  No.  6 — in 
No.  7 — in  No.  8.  Are  all  numbers  susceptible  of  exact  powers  and  roots  ?  9, 10. 
What  classification  is  made  in  reference  to  such  numbers?  11,  12.  Give  an 
example.  13. 


*  Evolution,  [c,  from, 
and  volvo,  to  roll.}  Tlie 
act  of  unfolding;  the  di- 
verse figures,  motions,  &c. 
of  a  body  of  soldiers.  Evo- 
lution is  so  called,  because 
the  root,  by  the  process, 
becomes  evolved  or  disen- 
tangled from  other  num- 
bers. 

20^ 


*  TABLE  OF  POWERS  AND  ROOTS. 

i 

1 

S3 

i 

1 

5 

i 
? 

£ 

si 

1 

c      1 

1 
2 
3 
4 

~5 
6 

~7 
~8 
~9 

1 

4 
~9 

16 
"25 

36 

49 
'64 

81 

1 

8 
27 
64 
125 
216 
343 
512 
729 

1 

16 

hi 

256 

625 

1296 

I 
32 

1 

1 

1 

1 

64 
729 

4090 

128 

256 

512 

243 
1024 
3125 

2187 

6561 

19683 
262144 

16384 

65536 

15625 

78125 

390()25 

1953125 
10077696 

7776 

46656 

279936 

1679616 

2401 

16807 

117649 

823543 

5764801 

40353607 
134217728 

4096 
6561 

32768 

262144 
531441 

2097152 

16777216 

59049 

4782969 

43046721 

387420489 

234  ARITHMETIC. 

ascertained,  because  it  is  done  by  multiplication,  which  produces  an 
exact  product. 

10.  On  the  contrary,  in  Evolution  there  are  many  numbers  whose 
roots  cannot  be  accurately  expressed,  as  the  square  root  of  2,  there 
being  no  factor  that,  multiplied  into  itself,  will  produce  it. 

1 1 .  Numbers  whose  roots  can  be  exactly  ascertained  are  called 
PERFECT  POWERS,  and  their  roots  rational  numbers. 

12.  But  other  numbers  are  called  imperfect  powers,  and  their 

roots  IRRATIONAL  NUMBERS,  Or  SURDS. 

13.  Thus  16  is  a  perfect  square,  because  its  root  is  a  rational 
number  ;  but  16  is  an  imperfect  cube,  because  there  is  no  factor  the 
third  power  of  which  is  that  number.     Its  root,  then,  is  a  surd. 

14.  By  the  means  of  decimals,  however,  we  can  come  nearer  and 
nearer  to  the  desired  root ;  that  is,  approximate  towards  it  to  any 
assignable  degree  of  exactness ;  as  the  square  root  of  2,  which  is 
nearly  1.41421356  +  . 

15.  Roots  are  often  indicated  after  the  manner  of  powers  in  Invp- 
lution,  the  numerators  of  which  show  the  powers  of  the  given  num- 
bers, and  the  denominators  the  required  roots  ;  thus — 

i  ± 

42  means  the  square  root  of  4^  or  4  ;  then  4^=2. 

I  1 

273  means  the  cube  root  of  27,  which  is  3;  then  273=3. 


16^  means  the  fourth  root  of  16,  which  is  2  ;  then  16*  =2. 


.4=, 


32  means  the  square  root  of  the  fourth  power  of  3  ;  then  32=9. 

6  6 

23  means  the  cube  root  of  the  sixth  power  of  2  ;  then  2^=4. 

16.  The  square  root  is  also  indicated  by  the  radical  sign  ■/»  and 
other  roots  by  placing  before  the  same  sign  their  respective  indices. 

17.  Thus,  -/O,  V8,  *-/16,  denote  the  square,  cube,  and  fourth 
raots,  respectively. 

18.  Since  -/ 25 =5,  therefore  ■/25X  v'25=25. 

19.  Since  V8=  2,  therefore  V8XV8X  V8=  8- 

20.  Since  *V  16=  2,  therefore  V 16  x  V 16  x  V 16  X  *^  16=  16. 

21.  But  V 16  X  V 16  X  V 16=  8,  since  *■/ 16=  2,  and  2  x  2  x  2  =  8. 

22.  When  numbers  have  a  line,  called  a  vinculum,  drawn  over 
them,  or  are  enclosed  in  a  parenthesis,  they  are  to  be  taken  together. 

23.  Thus,  3-/30-3  or  V(30-3)  means  that  3  is  first  to  be  taken 
from  30,  leaving  27,  of  which  the  cube  root  is  to  be  extracted. 


24.  Find  by  trial  the  difference  between  the  square  of  81  and  the 
square  root  of  81.  ^-  6,552. 

Q.  How  may  surd  roots  be  expressed  with  a  tolerable  degree  of  exactness?  14. 
How  are  roots  indicated  ?  15.  What  is  meant  by  42  ?  [Read  4  with  the  index  i] 

1_  I  jl  6 

What  by  273?— by  IG''?— by  32  ?— by  2^?  What  other  indications  of  roots 
are  there?  16.  Give  an  example.  See  17.  When  are  numbers  to  be  taken 
jointly?  22. 


EVOLUTION. 


235 


28.  What  is  the  amount  of  32',  32^,  and  V64  ?- 


1 

25.  Find  the  difference  between  125=*  and  125^ ;— between  16* 
and  V 16.  ^-  1,953,120;  65,534. 

26.  What  is  the  sum  of  3^  and  V 16 1— of  9^  and  V 16  x  V 16  X 

V16XV161  A.   13;  19. 

1 

27.  What  is  the  difference  between  04^  and  V641— between 

)|of  Jof54» 
and  V 2T  X  ;/25 !  A.  33,554,436  ;  787,35  ^. 

29.  Add  together  36^,V32,5*  Vi  W2  of  i  and  3/l0f-2.75. 

A.  636. 

30.  Suppose  an  orchard  has  2,500  trees,  and  that  it  is  in  the  form 
of  a  square  ;  how  many  trees  are  there  in  each  row "?      A.  50  trees. 

31.  A  man  desirous  of  appropriating  2^  acres  of  a  certain  lot  of 
land  for  a  vegetable  garden  in  the  form  of  a  square  ;  what  would  be 
the  distance  round  the  garden  1  A.  80  rods. 

32.  A  countryman  in  returning  from  market,  said  he  received  for 
his  butter  $4.41,  and  that  he  got  as  many  cents  a  pound  as  there  were 
pounds;  how  many  pounds  had  he,  and  what  was  the  price  per 
pound  1  A.  211b.  at  21  cents. 

33.  If  in  digging  a  cellar  of  equal  length,  breadth  and  depth,  there 
was  thrown  out  1,331  solid  feet,  how  deep  must  the  cellar  have 
been?  ^.11  feet. 

34.  Suppose  that  there  are  two  square  floors,  one  containing  121 
square  feet,  and  the  other  400 ;  now  what  is  the  sum  of  all  the  sides 
of  both  squares^  .A.  124  feet. 

35.  If  a  pile  of  wood  in  the  form  of  a  cube,  sold  at  $7.50  per  cord, 
comes  to  $3,750,  what  must  be  either  the  length,  breadth  or  depth  of 
the  pile]  ^.  40 feet. 

36.  The  foregoing  process,  when  the  numbers  are  large,  is  so  tedi- 
ous, that  rules  have  been  invented,  by  which  the  extraction  of  the 
required  root  is  rendered  comparatively  easy. 

EXTRACTION    OF    THE    SQUARE    ROOT. 
XCIX.     1.  The  Square  of  any  given  number,  is  the  product  of 
that  number  multiplied  by  itself     (xcvii.  3.) 

2.  The  Square  Root  of  any  given  number,  is  such  a  number,  as 
will,  on  being  multiplied  by  itself,  produce  the  given  number,  (xcviii.  2.) 

3.  A  square  figure  has,  as  we  have  seen,  four  equal  sides  and  four 
equal  angles,  (vii.  36).  The  length  of  a  square,  multiplied  by  its 
breadth,  produces  its  square  content  or  superficies,  sometimes  called 
its  area.       (vii.  46). 

4.  The  length  and  breadth  of  a  square  being  equal,  the  square  of 

XCIX.  Q.  What  is  a  square  ?  1.  What  is  the  square  root  ?  1.  Pescribe  a 
square  figure.  3.  What  is  meant  by  the  area  of  a  square  ?  3.  How  is  it  found  ? 
4.    Of  what  use  is  the  area  in  finding  the  side  of  a  square?  4. 


236 


ARITHMETIC. 


either  of  its  sides  is  equal  to  its  area ;  of  course  tiie  square  root  of  its 
area  is  equal  to  the  length  of  either  of  its  sides. 

6.  When  a  garden,  which  is  laid  out  in  the  form  of  a  square,  con- 
tains 1,296  square  rods,  what  is  the  length  of  each  side  ]  that  is,  what 
is  the  square  root  of  1296  I 

OPERATIONS. 


1st. 
Square  Rods. 
3  0)1296(3 
9  0  0 

6  0  +  6=6  6)3  9  6(6 
3  9  6 


2d. 
Square  Rods. 
3)1296(3 
9 


6  6)3  9  0 
3  9  6 


In  this  example,  we 
know  that  the  root  or 
the  length  of  one  side 
of  the  garden  must  be 
greater  than  30,  for  30^ 
=900,  and  less  than 40, 
for  402=:! GOO,  which 
is  greater  than  1296; 
therefore  we  take  30, 
the  less,  and,  for  con- 
venience' sake,  write 
it  at  the  left  of  1296, 
as  a  kind  of  divisor, 
likewise  at  the  right  of 
1296,  in  the  form  of  a 
quotient  in  division ; 
(See  Operation  1st. ;) 
then  subtracting  the 
square  of  30,  =900  sq 
rods,  from  1296  square 
rods,  leaves  396  square 
rods. 

The  pupil  will  bear 
in  mind  that  the  Fig 
on  the  left  hand  is  the 
form  of  the  garden  and 
contains  the  same  num- 
ber of  square  rods,  viz. 
1296.     This   figure  is 
divided  into  parts,  call- 
ed A,  B,  C,  and  D.    It 
will  be  perceived  that 
the  900  sq.  rds.  which 
we  deducted,  are  found 
by      multiplying     the 
length  of  A,  being  30 
rds.  by  the  breadth,  be- 
ing also  30  rods,  that 
is,  302=900. 
To  obtain  the  square  rods  in  B,  C,  and  D,  the  remaining  parts  of  the  figure, 
we  may  multiply  the  length  of  each  by  the  breadth  of  each,  thus  ;  30  x  6=180; 
CX6=36;  and  30x6=180;  then  180  +  36  +  180=396  square  rods;  or,  add  the 
length  of  B,  that  is,  30,  to  the  length  of  D,  which  is  also   30,  making  60  ;  or, 
which  is>  the  same  thing,  we  may  double  30,  making  60  ;   to  this  add  the  length 
of  C,  6  rods,  and  the  sum  is  66.     Now,  to  obtain  the  square  rods  in  the  whole 
length  o^B,  C,  and  D,  we  multiply  their  length,  6  rods,  by  the  breadth  of  each 
side,  th".s,  66  x  6=396  square  rods,  the  same  as  before. 

We  do  the  same  in  the  operation ;  that  is,  we  first  double  30  in  the  quotient, 
and  add  the  6  rods  to  the  sum,  making  66  for  a  divisor;  next,  multiply  66,  the 
divisor,  by  6  rods,  the  width,  making  396 ;  then  taking  396  from  396  leaves  0. 

The  pupil  will  perceive,  the  only  difference  between  the  1st  and  2d  opera- 
tion (which  see)  is,  that  in  the  2d  we  neglect  writing  the  ciphers  at  the  right  of 
the  numbers,  and  use  only  the  significant  figures.  Thus,  for  30  +  6,  we  write 
3  (tens,)  and  6  (^units,)  which,  joined  together,  make  36 ;  for  900,  we  write  9 
(hundreds).    This  is  obvious  from  the  fact,  that  the  9  retains  its  place  under 


30  rods. 

6  rods. 

o 

3  0 
B                6 

1  8  0 

6 
C     6 

3  6 

A 

Rods. 

3  0  ,     length  of  A. 
3  0  ,  breadth  of  A. 

D 

3  0 
6 

1  8  0 

O 
CO 

9  0  0  ,  sq.  rods  in  A. 

'§ 


30  rods. 


6  rods. 


SQUARE  ROOT.  237 

the  2  (hundreds).  Instead  of  60+6,  we  write  66.  Omitting  the  ciphers  in 
this  manner  cannot  possibly  make  any  difference,  and,  we  see,  it  does  not,  for 
the  result  is  the  same  in  both. 

6.  By  either  of  the  foregoing  operations,  then,  we  find  that  the  length  of  each 
side  of  the  garden  is  36  rods  ;  or,  that  the  square  root  of  1296  is  36. 

7.  Proof.    All  the  parts  of  the  above  figure  make  as  follows, — 
A  contains   9  0  0  square  rods. 

B  contains  18  0  square  rods.  Or;  by  Involution,  thus,  36  rods 

C  contains      3  6  square  rods.  X36  rods=1296  square  rods. 

D  contains    18  0  square  rods. 
The  given  sum  1  2  9  6^  square  rods. 

8.  If  then  the  square  of  the  root,  found  from  the  operation,  be 
equal  to  the  given  sum,  the  work  is  right. 

9.  Since  the  square  of  99,  the  greatest  factor  of  two  figures,  is 
9801,  which  has  the  same  number  of  figures  as  both  its  factors,  or 
only  double  the  number  of  figures  in  the  root  99,  therefore, — 

10.  The  square  of  any  root  cannot  have  more  figures  than  double 
the  number  of  figures  in  the  root. 

11.  Since  the  square  of  10,  the  least  factor  of  two  places,  is  100, 
which  has  only  one  figure  less  than  both  its  factors,  or  only  one  less 
than  double  the  number  of  figures  in  the  root,  therefore, — 

12.  The  square  of  any  root  can  never  have  but  one  figure  less 
than  double  the  figures  of  the  root. 

13.  Hence,  if  we  divide  any  given  number  into  periods  of  two  fig- 
ures each,  the  number  of  periods  will  equal  the  number  of  figures  of 
which  the  root  will  consist. 

RULE. 

14.  Point  off  the  given  number  into  periods  of  two  figures  each,  by 
puttting  a  dot  over  the  units,  another  over  the  hundreds,  and  so  on ; 
and  if  there  are  decimals,  point  them  in  the  same  manner,  from  the 
units  towards  the  right  hand. 

15.  Find  the  greatest  square  in  the  last  period  on  the  left,  write  its 
root  on  the  right,  as  a  quotient,  subtract  the  square  from  the  said  pe- 
riod, and  to  the  remainder  bring  down  the  next  period  for  a  dividend. 

16.  Double  the  root  {quotient)  for  a  partial  divisor,  and  on  its 
right,  place,  for  the  total  divisor,  such  a  figure  as  will  express  the 
greatest  number  of  times  that  the  true  divisor  is  contained  in  the 
dividend,  which  figure  will  be  the  second  in  the  root,  or  quotient. 

17.  Multiply  the  divisor  by  the  last  quotient  figure ;  subtract  the 
product  from  the  dividend;  and  to  the  remainder  bring  doivn  the  next 
period  for  a  new  dividend,  with  which  proceed  as  before,  by  doubling 
all  the  figures  in  the  quotient,  or  root,  6fC. 

Q.  How  is  the  operation  proved?  8.  What  is  the  greatest  number  of 
figures  which  any  root  can  have  1  10.  What  is  the  least  number  ?  12.  What 
reason  is  given  for  each?  9,  11.  What  is  the  inference?  13.  What  is  the 
rule  for  pointing  off  the  given  number  ?  14.  How  is  the  first  dividend  ob- 
tained? 15.  What  is  the  direction  for  finding  the  second  figure  in  the  root? 
16.  Whatfor  finding  the  next  dividend?  17.  Repeat  the  entire  rule.  14,15 
16,  17. 


238 


ARITHMETIC. 


18.  OPERATION.  Find  the  sq.  root  of  7569.  A.  87. 

8)7  5  6  9(8  7  Find  the  sq.  root  of  9025.  A.  95. 

_6_4  Find  the  sq.  root  of  4225.  A.  65. 

1  6  7)1  1  6  9  Findthesq.  root  of  1369.  JL.  37. 

^16  9  Find  the  sq.  root  of  2304.  A.  48. 

Proof.  8  7x8  7=7  5  6  9  Find  the  sq.  root  of  6561.  ^.81. 

19.  Recollect  to  double  all  the  quotient  figures  for  a  divisor. 

2)6553  6(256      "^^^^  *^®  ®^-  ^®°^  of  65536.     A.  256. 

4  Find  the  sq.  root  of  470596.  A.  686. 

4  5  )2  5  5  -^^"^  *^®  ^'l-  ^^^*  ^^  123201'.  A.  351. 

2  2  5  Find  the  sq.  root  of  801025.  A.  895. 

5  0  6)3  0  3  6  ^^^^  ^^®  ^^-  ^°°*  of  412164.  J..  642. 

3  0  3  6  Find  the  sq.  root  of  966289.  A.  983. 

==  Find  the  sq.  root  of  765625.  A.  875. 

20.  Extract  the  square  root  of  2 125764.  J..  1458. 

21.  Extract  the  square  root  of  6718464.  A.  2592. 

22.  Extract  the  square  root  of  4294967296.  4.65536. 

23.  When  the  divisor  is  too  large,  increase  the  dividend  by  bring- 
ing down  the  next  period  of  the  given  sum,  then  place  a  cipher  in  the 
root,  and  find  a  new  divisor  as  before. 

2)4202  5(205     Find  the  sq.  root  of  42025.  A.  205. 

4 Find  the  sq.  root  of  651249.        A.  807. 

4  0  5)2025  Find  the  sq.  root  of  49126081.  A.  7009. 

2  0  2  5  Find  the  sq.  root  of  25806400.  A.  5080. 

24.  What  is  the  square  root  of  6480.25?  [See  R.  14.]  A.  80.5. 
8)64  8  0.2  5(8  0.5  Find  the  sq.  root  of  913.8529.    .A.  30.23. 

^  4 Findthesq.  root  of  9. 3025.  A.  3.05. 

1605)8025  Findthesq.  root  of  .00015625.  A.  .0125. 

8  0  2  5  Find  the  sq.  root  of  196.5604.    A.  14.02. 

25.  Extract  the  square  root  of  .0000000001018081.  A.  .00001009. 

26.  When  the  last  divisor  leaves  a  remainder,  the  operation  may 
3)10(3.16+     be  continued  by  annexing  successively  peri- 
ods of  decimal  ciphers. 
Find  the  sq.  root  of  10.  A.  3.162  + 
Find  the  sq.  root  of  175.        A.  13.228+ 
Find  the  sq.  root  of  90.         A.  9.4868  + 
Find  the  sq.  root  of  5.          A.  2.23606  + 
Findthesq.  rt.  of  2.  A.  1.41421356237  + 

27.  When  the  last  period  of  a  decimal  consists  of  only  one  figure, 
annex  a  cipher  to  complete  the  period. 

•    28.  What  is  the  square  root  of  11.7 1  A.  3.4205+ 

29.  What  is  the  square  root  of  8.003 1  A,  2.828+ 

30.  What  is  the  square  root  of  .018?  A.  .1341  + 

Q.  What  is  the  direction  when  the  divisor  is  not  contained  in  the  dividend  ? 
23.  What  is  to  be  done  with  the  final  remainder  ?  26.  What  with  an  imperfect 
decimal  period  ?  27. 


6  1)1. 

0 
6 

0 

1 

6  2  6)3 

3 

smainder. 

9 

7, 

.0 
.  5 
.  4 

0 
6 
4 

SQiJARE   ROOT.  239 

§1.  When  fractions  have  terms  that  are  perfect  powers,  [xcviii.  11  ] 
Extract  the  roots  of  the  most  simple  terms. 

32.  What  is  the  square  root  of  rVr  •  ^-  T2  or  |- 

33.  What  is  the  square  root  of  ^Vo  ^  -^-  M- 

34.  What  is  the  square  root  o^YihHTT^  ^-  TffT- 

35.  When  the  terms  are  either  of  them  imperfect  powers,  [xcviii. 
12.]  reduce  them  first  to  a  decimal. 

36.  What  is  the  square  root  of  ^^  A.  .9128  + 

37.  What  is  the  square  root  of  |^  1  A.  .9198  + 

38.  What  is  the  square  root  of  ^  ?  A.  .83205. 

39.  What  is  the  square  root  of  |  ?  A.  .80(3  -h 

40.  What  is  the  square  root  of  i?   f?    |?  |?  ^?   4?  ^1   ^^ 
A.  .707+  ;.816+  ;.894+  ;.912+  ;  i;.755+  ;.745+  ;.577  +  . 

41.  A  mixed  number  may  first  be  reduced  to  an  improper  fraction, 
and  its  roots  be  expressed  again  by  a  mixed  number,  unless  its  terms 
are  imperfect  powers,  in  which  case  the  operation  must  be  conducted 
decimally. 

42.  Extract  the  square  root  of  420^^.  ^.20^ 

43.  Extract  the  square  root  of  91227'  A.  30^. 

44.  Extract  the  square  root  of  272  j.  A.  16|^. 

45.  What  is  the  square  root  of  Hy^^l  .A.  4.1509+ 

46.  What  is  the  square  root  of  87^V2  •  ^-  ^.35  + 

47.  What  is  the  root  indicated  by  V234y\1  A.  15. 3 196  + 

48.  What  is  the  root  indicated  by  81y*f y  1  A.  9.000862  + 

49.  If  the  next  divisor  or  double  of  the  root  be  written  under  the 
final  remainder,  the  fraction  will  express  very  nearly  the  radical  re- 
mainder, which  should  be  first  reduced,  either  to  its  lowest  terms,  or 
to  a  decimal,  and  annexed  to  the  root.* 

50.  How"  much  is  -/ 10946 1  =  104  and  130  rem.:  104x2^203, 
the  next  divisor.  A.  104if^  =  104f  or  104.625. 

51.  How  much  is./43256789101 1  A.  207982t^Ul' 

52.  How  much  is  ^V 54301 0940567 1  A.  736892|lflf  |f . 

53.  How  much  is  ■/ 6732100100954 !  A.  2594629 /yV^J^. 

64.  How  much  is -/ 1000900010007  1  A.  IQOOUd^UUll 

65.  From  ■/729  take  1442+22.  A.  11. 

66.  From  ^729  take  256^+22.  A.  7. 

67.  From  20^  take  a/46311.04.  A.  7784.8. 

58.  From  4^  take  v/ 9 15^1^+1. 10252.  A.. 7. 

59.  How  much  are  v/ 529  +  ^1764 +144^+ 14592642 +  ^67241 

A.  1367. 

Q.  What  is  the  rule  for  extracting  the  roots  of  fractions?  31,  35.  What  for 
mixed  numbers  ?  41. 

*  Although  the  remainder  is  a  little  too  great  in  the  square  root,  and  a  little  too  small 
in  the  cube  root,  they  are  nevertheless  sufficiently  exact  for  most  purposes,  and  much 
more  convenient  than  the  operation  by  annexing  ciphers. 


340  ARITHMETIC. 


60.  From  ;/ 152399025  take  (v'4120900+y/y+.00060025\) 

A.  10314.7255. 

61.  What  is  the  square  root  of  15241578750190521 1 

A.  The  9  digits. 

62.  Find  the  sum  of  the  roots  or  numbers  involved  in  all  the  per  • 
feet  squares  between  1  and  100.  A.  44. 

63.  Find  the  sum  of  the  squares  whose  roots  are  surds,  between  1 
and  20?  A.  160. 

64.  Suppose  that  a  commandant  of  an  army  has  180625  effective 
men,  and  would  form  them  into  a  solid  square,  how  many  would 
there  be  in  each  rank  and  file  ]  A.  -/ 180625=425. 

65.  Suppose  a  town  proposes  to  levy  a  poll  tax  of  8216.09  so  that 
each  man  shall  pay  as  many  cents  as  there  are  men  to  be  taxed ; 
what  is  each  man's  tax  on  his  head  ?  A.  $1.47. 

66.  Suppose  there  are  two  portions  of  land  each  in  the  form  of  a 
square,  and  that  one  is  SO}  miles  square,  and  the  other  contains  30y 
square  miles ;  what  is  the^um  of  the  distances  round  both  squares? 

A.  143  miles. 

67.  If  the  surface  of  the  earth,  which  is  computed  to  contain  196.- 
000,000  square  miles  were  in  the  form  of  a  square,  what  would  be  the 
distance  round  it  1  A.  56,000  miles. 

68.  If  a  tract  of  land  6}  miles  long,  and  4  miles  wide,  which  cost 
^Ij  per  acre  be  exchanged  for  the  same  quantity  in  the  form  of  a 
square,  and  subsequently  be  divided  into  one  hundred  equal  and  square 
farms,  i  of  which  should  bring  at  auction  $llf  per  acre  ;  |  of  them 
$12  per  acre,  and  the  rest  $10j  per  acre  ;  what  would  be  the  profit 
in  the  transaction,  and  what  the  sum  of  the  distances  round  all  the 
squares  ?  A.  200  miles  ;  $164020  profit. 

PROPORTIONS     INVOLVING    ROOTS    AND    POWERS. 

69.  The  product  of  the  square  roots  of  any  two  numbers,  is  equal 
to  the  square  root  of  their  product.  -  * 

70.  Prove  -/Six  v^ 22 5=/ 81x225. 

71.  To  find  a  mean  proportional  between  any  two  numbers: — 
Extract  the  square  root  of  their  product. 

72.  For  in  the  proportion  2  :  10  :  :  10  :  50 ;  of  which  the  10  is  a 
mean  proportional  between  2  and  50 ;  we  have  on  geometrical  princi- 
ples, 2  x  50=1 0^ 

73.  What  is  the  mean  proportional  between  3  and  12?  4  and  36? 
24  and  96?   16  and  64?  ^.  6  ;  12  ;  48  ;  32. 

74.  What  is  the  mean  proportional  between  25  and  289  ?  25  and 
156.25?  A.  85;  62|. 

75.  What  is  the  mean  proportional  between  7  and  If  ?  10^  and 
4U?  A.  3h;  20.6. 

Q.  To  what  is  the  product  of  the  square  root  of  any  two  numbers  equal  ?  69. 
How  is  a  mean  proportional  between  any  two  numbers  found?  71.  What  is 
the  mean  proportional  between  4  and  9  ?  between  2  aini  18? 


EXTRACTION  OF  THE  CUBE  ROOT.  241 

76.  The  mean  proportional  between  any  two  numbers,  has  the  same 
ratio  to  those  numbers,  that  the  square  roots  of  those  numbers  have 
to  each  other. 

77.  Find  the  mean  proportional  between  25  and  36,  and  the  ratio 
between  it  and  those  numbers,  and  see  if  it  is  the  same  as  the  ratio 
between  -/25  :  -/36. 

78.  *  To  find  any  two  numbers  from  having  their  sum  and  product 
given  :-^From  the  square  of  half  their  sum,  subtract  their  product  i 
extract  the  square  root  of  the  remainder,  and  add  it  to  half  their  sum^ 
for  the  larger  number ;  or  subtract  it  therefrom  for  the  smaller  number. 

79.  A  certain  field  contains  an  area  of  30  acres  2  roods  and  20. 
rods :  required  its  length  and  breadth,  the  sum  of  these  being  148 
rods.  A.  98rd.  :  50rd. 

80.  A  gentleman  having  purchased  a  certain  quantity  of  flour,  for 
$1935,  found  that  if  he  added  the  number  of  dollars  it  cost  per  barrel 
to  the  number  of  barrels,  the  sum  would  be  224.  IIow  many  barrels 
must  he  have  bouglit  1  A.  215  barrels. 

81.  To  find  any  two  numbers  from  having  their  sum  and  the  sum 
of  their  squares  given  : — Find  the  difference  between  the  square  of 
their  sum,  and  the  su?n  of  their  squares :  half  this  difference  subtract 
from  the  square  of  half  their  sum,  and  add  the  square  root  of  the  re- 
mainder to  their  half  sum  for  the  greater  number,  or  subtract  it  there' 
from  for  the  smaller  number. 

82.  Suppose  that  two  square  fields  contain  together  9A.  2R.  5rd. 
and  that  the  sum  of  either  their  length  or  breadth  is  55  rods  ;  pray 
what  is  the  length  of  each  lot  ]  A.  25rd.  :  30rd. 

EXTRACTION  OF  THE  CUBE  ROOT. 

C.  I.  The  Cube  of  any  given  number  is  the  product  of  that  number 
multiplied  by  its  square,     [xcvii.  3.] 

2.  The  Cube  Root  of  any  given  number,  is  such  a  number  as  will, 
on  being  multiplied  by  its  square,  produce  the  given  number, 
txcviii.  2.] 

3.  A  body  in  the  form  of  a  cube  is  a  solid  of  six  equal  sides,  each 
containing  an  exact  square.  [See  the  block  accompanying  this  work.] 

4.  A  Cube  then  has  three  dimensions,  viz.,  length,  breadth,  and 
thickness  or  depth  ;  the  product  of  which  multiplied  into  eaeh  other  is 
called  its  solid  content,     [vn.  60.] 

5.  The  length,  breadth,  and  thickness  of  a  cube  being  equal,  the 
cube  of  either  of  its  sides  must  be  equal  to  its  solid  contents ;  of 
course  the  cube  root  of  its  solid  contents  must  be  equal  to  the  length 
of  either  of  its  sides. 

Q.  To  what  is  the  ratio  of  any  mean  proportional  equal?  76.  How  are  any 
two  numbers  found  from  having  their  sum  and  product  given  ?  78. — -from  hav- 
ing their  sum  and  the  sum  of  their  squares  given?  81. 

C  Q.  1.  What  is  the  cube  of  any  number?  1.  What  is  the  cube  root?  2. 
What  is  a  cubical  body  ?  3.  What  its  dimensions?  4.  How  are  its  solid  con- 
tents found  ?  5.     How  either  of  its  dimensions  ?  5. 

*  This  and  the  foUowing  proportion  are  deduced  from  Algebraic  processes 
21 


242  ARITHMETIC. 

6.*  The  blocks  which  accompany  this  work  for  the  purpose  of  il- 
lustrating the  operation  of  the  following  example  are  eight  in  all,  and 
when  put  together,  they  should  form  a  perfect  cube  of  24,389  sd.  feet. 

7.  These  blocks  are  marked  by  the  letters  A,  B,  C,  and  D,  whose 
proportional  dimensions  are  supposed  to  be  as  follows  : 

A  is  a  cube,  20  feet  long,  20  feet  wide,  and  20  feet  thick. 
Three  B's,  each  20  feet  long,  20  feet  wide,  and  9  feet  thick. 
Three  C's  each  20  feet  long,  9  feet  wide,  and  9  feet  thick. 
D  is  a  cube  9  feet  long,  9  feet  wide,  and  9  feet  thick. 

8.  If  a  cubical  block  which  is  formed  by  the  8  small  ones  above, 
.contains  24,389  solid  ft. ;  what  must  be  the  length  of  each  of  its  sides? 

243  89(20  In  this  example, 

2  0    -a  0  00  side  cannot  be  30ft., 

2^X  300  =  120  0)16389  (  9  for  303=27000  solid 

^^■        ,  ^>AAv^,l/^    o—  t  n  Q  n  n  ^^^^^  a,ve    more  than 

DlV.    1  2  0  0  X  quo.  9-10800  24389,  the  given  sum 

2X30X      9-48C0  -therefore,  we  ^vill 

^=        7  2  9  take  20  for  the  length 

16  3  8  9  ^^  ^^*^  ^^^®  °^  ^^^ 

The  sajne  without  the  ciphers  ^"^"i;,^  20  x  20  x  20 

2  4  3  8  9(29  Ans.  ^gooo   solid    feet. 

^        " which  we  must,  of 

22X300  =  1200)  16  3  8  9  dividend.  course,  deduct  from 

12  0  0X9=     10800  24389  leaving  16389. 

2X3  0X9J=         4800  These  8000  solid 

'  ^  ^  feet  the   pupil   will 

16  3  8  9   subtrahend,    perceive,    are    the 

~  solid  contents  of  the 


cubical  block  marked  A.  This  corresponds  with  the  operation  ;  for  we  write 
20  feet,  the  length  of  the  cube  A,  at  the  right  of  24389,  in  the  form  of  a  quo- 
tient ;  and  its  square  8000,  under  24389 ;  from  which  subtracting  8000,  leaves 
16389  as  before. 

As  we  have  16389  cubic  feet  remaining,  we  find  the  sides  of  the  cube  A  are 
not  so  long  as  they  ought  to  be ;  consequently  we  must  enlarge  A ;  but  in  doing 
this  we  must  enlarge  three  sides  of  A,  in  order  that  we  may  preserve  the  cu- 
bical form  of  the  block.  We  will  now  place  the  three  blocks  each  of  which  is 
marked  B,  on  these  three  sides  of  A.  Each  of  these  blocks,  in  order  to  fit,  must 
be  as  long  and  as  wide  as  A;  and,  by  examining  them,  you  will  see  that  this  is 
the  case  ;  that  is,  they  are  20  feel  long  and  20'^feet  wide  ;  then  20X20=400, 
the  square  contents  in  one  B  ;  and  3  x  400=1200,  square  contents  in  three  Bs ; 
then  it  is  plain,  that  16389  solid  contents,  divided  by  1200,  the  sq.  contents  will 
give  the  thickness  of  each  block.  But  an  easier  method  is  to  square  the  2,  (tens,) 
in  the  root  20,  making  4,  and  multiply  the  product  4,  by  300,  making  1200,  a  divi- 
sor, the  same  as  before. 

Wedo  the  same  in  the  operation  (which  see);  we  multiply  the   square  of  the 

*This  rule  is  best  illustrated  by  means  of  blocks  which  may  be  supposed  to  contain  a 
certain  proportional  number  of  feet,  inches,  «fcc.,  corresponding  with  the  operation  of 
the  rule.  They  may  be  made  in  a  few  minutes,  from  a  small  strip  of  pine  board,  with 
a  common  penknife,  at  the  longest,  in  less  time  than  the  teacher  can  make  the  pupil 
comprehend  the  reason,  from  merely  seeing  the  picture  on  paper.  This  method  of  de- 
monstrating the  rule  will  be  an  amusing  and  instructive  exercise,  both  to  teacher  and 
pupil,  and  may  be  comprehended  by  any  pupil,  however  young,  who  is  so  fortunate  as 
to  have  progressed  as  far  as  this  rule.  It  will  give  him  distinct  ideas  resjHicling  the 
diflferent  dimensions  of  square  and  cubic  measures,  and  indelibly  fix  on  his  mind  the 
reason  of  the  rule,  and  consequently  the  rule  itself.  But,  for  the  convenience  of  teach- 
ers, blocks  illustrative  of  the  operation  of  the  foregoing  example,  accompany  this  work 


CUBE    ROOT  243 

quotient  figure,  2,  l)y  300,  thus  2  x  2=4x300=1200  ;  then  the  divisor,  1200 
(the  square  contents)  is  contained  in  1C3S9  (solid  contents)  9  times,  that  is,  Oft. 
is  the  thickness  of  each  bloc!;  marked  B.  This  quotient  fyiurc,  9,  we  place  at  the 
right  of  16389,  and  then  1200  square  feet  x  9  feet,  the   thickness,=  10800  s.  ft. 

If  we  now  examine  the  block,  thus  increased  by  the  addition  of  the  3  Bs,  we 
shall  see  that  there  are  yet  three  corners  not  filled  up  ;  these  are  represented  by 
the  three  blocks,  each  marked  C,  and  each  of  which,  you  will  perceive,  is  as  long 
as  either  of  the  Bs,  that  is,  20  ft.,  being  the  length  of  A,  which  is  20  in  the  quo- 
tient. Their  thickness  ami  breadth  are  the  same  as  the  thickness  of  the  Bs, 
which  we  found  by  dividing,  to  be  9  feet,  the  last  quotient  figure.  Now,  to  get 
the  solid  contents  of  each  of  these  Cs,  we  multiply  their  thickness  (9  feet)  by 
their  breadlh,  (9  feet,)=81  sc^uare  feet;  that  is,  the  square  of  the  last  quotient 
figure,  9=81  ;  these  scjuare  contents  must  be  multiplied  by  the  length  of  each, 
(20  feet,)  or,  as  there  are  3,  by  3x20=60;  or,  which  is  easier  in  practice,  we 
may  multiply  the  2,  (tens)  in  the  root,  20,  by  30,  making  GO,  and  this  product  by 
92=81,  tlie  square  contents  =  48C0  solid  feet. 

We  do  the  same  in  the  operation,  by  multiplying  the  2  in  20  by  30=60  X  9  X 
9=4860  solid  feet,  as  before;  this  4960  we  write  under  the  10800,  for  we  must 
add  the  several  products  together  by  and  by,  to  know  if  our  cube  will  contain 
all  the  retiuircd  feet. 

By  turning  over  the  block  with  all  the  additions  of  the  blocks  marked  B  and 
C,  which  are  now  made  to  A,  we  shall  spy  a  little  scjuare  space,  which  prevents 
the  figure  from  becoming  a  complete  cube.  The  little  block  for  this  comer  is 
marked  D,  which  the  pupil  will  find,  by  fitting  it  iu;  to  exactly  fill  up  this  space. 
This  block  D,  is  exactly  square,  and  its  lengi.h,  breadih  and  thickness  are  alike, 
and,  of  course,  equal  to  the  thickness  and  width  of  the  Cs,  that  is,  9  feet,  the 
last  quotient  figure  ;  hence  9ft.  X  Oft.  X  9ft.=729  solid  feet  in  the  block  D  ;  or,  in 
other  words,  the  cube  of  9,  (the  (piotient  figure,)  which  is  the  same  as  93=729, 
as  in  the  operation.  We  now  write  the  729  under  the  4860,  that  this  may  be 
reckoned  in  with  the  other  additions. 

We  next  proceed  to  add  the  solid  contents  of  the  Bs,  Cs,  and  D,  together, 
thus,  10800x4860x729=16389,  precisely  the  number  of  sulid  feet  which  we 
had  remaining  after  we  deducted  6000  feet,  the  solid  contents  of  the  cube  A. 

If,  in  the  operation,  we  subtract  the  am*int,  10389,  from  the  remainder,  or 
dividend,  16389,  we  shall  see  that  oiir  additions  have  taken  all  that  remained, 
after  the  first  cul)e  vias  deducted,  there  being  no  remainder. 

The  last  little  block,  when  fitted  in,  as  you  saw,  rendered  the  cube  complete, 
each  side  of  which  we  have  now  Ibund  to  be  20-f  9=29  feet  long,  which  is  the 
cube  root  of  24389  (solid  feet) ;  but  let  us  see  if  our  cube  contains  the  retpiired 
number  of  solid  feet. 

9.  Proof.— 8000  s.ft.  in  A-flOSOO  s.  ft.  in  3  Bs  +  4860  s.  ft.  in  3  Csx729  s. 
ft.  in  D=24389  s.  ft.  in  the  given  sum  which  because  they  are  equal  to  29^  form 
a  perfect  cube,  then,  29  is   the  length  of   the  recjuired  side  ;  therefore, — 

10.  If  by  Involution  the  cube  of  the  root  found  from  the  operation  be  e<^ual  to 
the  given  sum,  the  operation  is  correctly  {jorformed. 

11.  By  reasoning  similar  to  that  employed  in  xcix.  9,  it  may  be 
shown  that  the  product  of  any  three  numbers  into  each  other  never 
has  more  figures  than  all  its  factors,  nor  fewer  than  that  same  num- 
ber less  two. 

12.  We  infer  also  from  the  same  reasoning,  that  if  we  point  off  any 
sum  into  periods  of  three  figures  each,  the  number  of  periods  will  equal 
the  number  of  figures  in  its  root.     Hence  the  direction  in  the  rule. 

RULE. 

13.  Divide  the  giucn  number  into  periods  of  three  Jigures  each,  by 
placing  a  point  over  the  unit  figure,  and  over  evert/  third  one  from  the 
place  of  units  to  the  left  in  whole  numbers,  and  to  the  right  in  decimals. 

Q,  Of  how  many  figures  will  every  root  eon.sist?  12.  What  is  the  reason  for 
it?  11.     VVhaT.  is  :he  rule  for  pointing  off  the  given  number  ?  13. 


244 


ARITHMETIC. 


14.  Find  the  greatest  cube  in  the  first  left  hand  period,  and  place 
its  root  in  the  quotient.  Subtract  the  cube  thus  found  from  this  peri- 
od, and  to  the  remainder  bring  down  the  next  period,  and  the  result 
will  be  the  dividend. 

15.  Multiply  the  square  of  the  root  or  quotient  by  300  for  a  divisor. 
Divide  the  dividend  by  the  divisor  for  the  next  figure  in  the  root. 

16.  Multiply  the  divisor  by  the  quotient  fig^u-e ;  multiply  the  former 
quotient  figure  or  figures  by  30  times  the  square  of  the  last  quotient 
figure;  finally,  cube  the  last  quotient  figure ;  then  add  these  three 
results  together  for  a  subtrahend. 

17.  Subtract  the  subtrahend  from  the  dividend,  and  to  the  remain- 
der bring  doion  the  next  period  for  a  new  dividend,  loith  which  pro- 
ceed as  before,  and  so  on  till  all  the  periods  are  brought  doum.* 

18.  Note.  When  the  subtrahend  happens  to  be  larger  than  the  dividend,  the 
quotient  figure  must  be  made  one  le.ss,  and  we  must  find  a  new  subtrahend. 
The  reason  why  the  quotient  figure  will  be  sometimes  too  large,  is,  because 
this  quotient  figure  merely  shows  the  width  of  the  three  first  additions  to  the 
original  cube ;  consequently,  when  the  subsequent  additions  are  made,  the 
width  (quotient  figure)  may  make  the  solid  contents  of  all  the  additions  more 
than  the  cubic  feet  in  the  dividend,  which  remain  after  the  solid  contents  of  the 
original  cube  are  deducted. 

19.  When  we  have  a  remainder  after  all  the  periods  are  brought  down,  we 
may  continue  the  operation  by  annexing  periods  of  ciphers,  as  in  the  square 
root.     When  it  happens  that  the  divisor  is  not  contained  in  the  dividend,  a  ci 


pher  must  be  written  in  the  quotient  (root,)  and 
bringing  down  the  next  period  in  the  given  sum. 


9663597(213,  Ans. 

2^  =  8 

2^X300^1200)1663  dividend. 
1200X1=  1200 
2x30xprz      60 

V= 1 

1261  subtrahend. 
X  300=132300)402597  dividend. 


21 


132300X3=396900 

21  X  30  X  32=     5670 

3=*=         27 


402597  subtrahend. 


a  new  dividend  formed  by 

20.  What  is  the  cube 
root  of  9663597?  A.  213. 

21.  What  is  the  cube 
root  of  91 125  1      A.  45. 

22.  What  is  the  cube 
root  of  970299]     A.  99. 

23.  What  is  the  cube 
root  of  778688  ?     A.  92. 

24.  What  is  the  cube 
root  of  2000376?  A.  126. 

25.  Wliat  is  the  cube 
root  of  3796116^  A.  156. 

20.  What  is  the  cube 
rootof94818816?  A456. 

27.  What  is  the  cube 
root  of  175616000?  560. 


Q.  What  for  finding  the  first  dividend  ?  14.  What  for  finding  the  subtra- 
hend? 16.  Describe  the  rest  of  the  process.  17.  What  is  the  whole  rule  ?  13, 
14,  15,  16,  17.  What  is  to  be  done  when  the  subtrahend  is  too  large  ?  18. 
What  when  the  divisor  is  too  large?  19.  What  is  to  be  done  with  the  final  re- 
mainder? 19. 


*  The  root  of  the  first  period  take. 
And  of  that  root  a  quotient  make  : 
Which  root  must  now  a  cube  become, 
To  be  a  period  taken  from ; 
To  the  remainder  then  you  must 
Bring  down  another  period  just ; 
Which  being  done,  you  then  must  see, 
Tills  number  straight  divided  be, 


By  just  three  hundred  times  the  square 
Of  what  the  quotient  figures  are  ; 
The  last  squared,  multiplied  by  the  rest, 
The  product  thirty  times  cxi)rest ; 
The  cube  of  the  last  figure,  too, 
You  must  put  in,  if  right,  you  do ; 
Add  these,  subtract  ihem ;  so  descend, 
From  point  to  point  unto  the  eiid. 


CUBE    ROOT.  245 

28.  What  is  the  cube  root  of  1,879,080,904 1  A.  1,234. 

29.  Where  the  divisor  is  larger  than  the  dividend.     [See  19.] 

30.  What  is  the  cube 


748613312(908 
729 


90»  X  300=  2430000)  1 9GI3312 

2430000  X  8  =19440000 

90X30X8'=     172800 

8'=___513 

19613312 


root  of  748,613,312? 
A.  908. 

31.  What  is  the  cube 
root  of  8,365,427? 

A.  203. 

32.  What  is  the  cube 
root  of  517,781,027? 

A.  803. 

33.  What  is  the  cube  root  of  731,189,187,729?  A.  9,009. 

34.  Find  the  cube  root  of  8,096,384,512,000,000,000  ? 

A.  2008000. 

35.  What  is  the  cube  root  of  .000,015,625  ?  A.  .025. 

36.  What  is  the  cube  root  of  12.167  ?  A.  2.3. 

37.  What  is  the  cube  root  of  26.2  ?  [See  xcix.  27.     A.  2.97-  + 

38.  What  is  the  cube  root  of  15.32  ?  A.  2.483  + 

39.  What  is  the  cube  root  of  fl?     [See  xcix.  31.]  A.  f 

40.  What  is  the  cube  root  of  U4  ^  ^-  t- 

41.  What  is  the  cube  root  of  ^|-^-  ?  A.  |}. 

42.  What  is  the  cube  root  of  ^j^g-?     [See  xcix.  35.41.]  A.  .13"+ 

43.  What  is  the  cube  root  of  ^VV?   mVWtt^  49  ^^  ?  7,5583^1? 

44.  What  is  the  cube  root  of  15f  ?  A.  2k- 

45.  What  is  the  cube  root  of  1,242}^?  A.  10|. 

46.  What  is  the  cube  root  of  l,984f^?  A.   12566  +  . 

47.  What  is  the  cube  root  of  200|75  ?  A.  5.859  +  . 

48.  AVliat  is  the  cube  root  of  183,457?  ?  A.  56.82  +  . 

49.  Howmuch  is  V 8,000 +  1,728 J?  A.  32. 

50.  What  is  132,651^^-3^]  A.  42. 

51.  Add  into  one  sum  the  cube  roots  of  274,625;  2,197;  6,859 

A.  97. 

52.  How  much  are  V4,096  +  l,OOo'^+V166|+  5123+.V9|f|? 

A.  41|.      ■ 

53.  To  find  two  mean  proportionals  between  any  two  given  num- 
bers, as  the  30  and  150  in  6  :  30  :  :  150  :  750:  in  which  5  is  the 
common  ratio.  Now  6x5^  =  750;  of  course  750-^-6  and  this  quo- 
tient by  125  =  5^  =  5  the  ratio  again,  then  6x5  the  ratio  =  30  the 
smaller  mean,  and  750-^-5  =  150  the  greater  mean  ;  therefore — 

54.  Divide  the  greater  of  the  two  numbers  by  the  smaller,  and  eX' 
tract  the  cube  root  of  the  quotient  for  the  common  ratio,  loith  which 

Q.  What  is  the  method  of  procedure  with  an  imperfect  decimal  period  ?  37. 
With  a  fraction?  39,  42.  With  a  mixed  number?  42.  What  is  the  rule' for 
finding  the  mean  proportionals  between  two  numbers  ?  54.  What  is  the  illus- 
tration? 55. 

21* 


246  ARITHMETIC. 

multiply  the  smaller  of  the  given  numbers  for  the  smaller  mean  prO' 
poriional,  and  divide  the  greater  of  the  given  numbers  by  the  same 
ratio  for  the  greater  mean  proportional. 

55.  What  are  the  two  mean  proportionals  between  32  and  16,384 1 

A.  256  :  2,048. 

56.  What  are  the  two  mean  proportionals  between  to  a^nd  37|  1 — 
between 'I  of  ^^  and  63  times  I  of  f  1      A.   1.5  and  1\  ;  yV  ^T^^  J- 

57.  What  are  the  two  mean  proportionals  between  .000625  and 
625?  A.  yVand6l. 

58.  Find  the  sum  of  the  cube  roots  of  all  the  numbers  under  1,001, 
which  are  perfect  powers  of  those  ro^tis.  A-  55. 

59.  Find  the  sum  of  all  the  poiccrs  under  20,  the  cube  roots  of 
which  are  surds.  A.  181. 

60.  If  the  amount  of  a  certain  sum  at  compound  interest  for  3 
years  be  Si.  191016  ;  w^hat  is  the  amount  for  the  first  year,  the  rate 
per  cent.,  and  the  principal?  A.  $1.06  ;  rate  6;  $1  principal. 

61.  Suppose  that  in  making  an  excavation,  there  were  throw^n  out 
838,561,807  solid  feet  of  earth;  what  would  be  the  length  of  one 
side  of  a  cube  of  equal  contents?  A.  943  feet. 

62.  If  a  pile  of  wood  which  is  2,565ft.  long,  40ft.  wide,  and  40ft. 
high,  be  thrown  into  the  form  of  a  cube  and  sold  for  ^  of  as  many 
dollars  as  the  cube  would  be  feet  long,  what  sum  would  the  cube 
bring?  A.  $96. 

63.  If  5,375  tons  of  round  stones  of  equal  size,  1,849  of  which 
just  weigh  one  ton,  be  thrown  into  a  cubical  pile,  and  all  be  sold  for 
what  the  number  of  stones  that  will  reach  across  one  side  of  the  cube 
would  bring  at  the  rate  of  $5  for  7  stones ;  what  would  be  the  pur- 
chase price?  A.  $153.57^. 

EXTRACTION    OF    THE    ROOTS    OF    ALL    POWERS. 

CI.  1.  When  the  index  of  the  given  power  is  a  composite  num- 
ber— Resolve  it  into  as  many  indices  or  factors  as  is  possible ;  then 
extract  the  roots  of  the  given  power  successively  as  their  indices  re- 
quire. 

2.  That  is,  extract  the  root  denoted  by  one  index,  then  the  root  of 
that  root,  as  denoted  by  another  index,  and  so  on  till  the  number  of 

•^  extraction's  shall  equal  the  number  of  indices. 

3.  Thus  the  4th  root  (2x2)  =  the  square  root  of  the  square  root ; 
the  sixth  root  (3x2)  =  the  cube  root  of  the  square  root,  or  the  square 
root  of  the  cube  root. 

4.  The  8th  root  (2x2x2)  =  the  square  root  of  the  square  root  ot 
the  square  root ;  the  ninth  root  (3x3)  ==  the  cube  root  of  the  cube 
root ;  the  10th  root,  (5x2,)  the  fifth  root  of  the  square  root,  «&c. 

5.  What  is  the  biquadrate  or  fourth  root  of  20,736  ?  ■/ 20,736  = 
144,  and  y/ 144=^12. ^ A.  12. 

CI.  Q.  How  are  the  roots  of  most  powers  extracted?  1.  What  is  meant  by 
that  process  ?  2.  How,  for  example,  are  the  fourth  and  sixth  roots  extracted  ?  3 
How,  the  8th  and  9th  roots?  4 


CUBE   ROOT.  247 

6.  What  is  the  biquadrate  root  of  2,998,219,536?  A.  234. 

7.  What  is  the  sixth  root  of   1,178,420,166,015,625  ?  A.  325. 

8.  What  is  the  eighth  rgot  of  722,204,136,308,736  ?  A.  72. 

9.  What  is  the  ninth  root  of  387,420,489 1  A.  9. 
10.  What  is  the  twelfth  root  of  282,429,536,481  ?  A.  9. 

GENERAL  RULE  FOR  EXTRACTING  ALL  ROOTS. 

11.  Point  off,  from  the  unWs  place.,  the  periods,  as  the  required 
root  directs ;  that  is,  for  the  fourth  root  point  off  periods  of  four 
figures  each ;  for  the  fifth  root,  periods  of  five  figures,  <5fC. 

12.  Find  by  trial  the  greatest  root  in  the  left  hand  period,  and  sub- 
tract its  power  from  the  said  period. 

13.  To  the  remainder  bring  down  the  next  figure  in  the  next  period, 
for  a  dividend. 

14.  Involve  the  root  to  the  power  next  inferior  to  that  which  is 
given,  and  multiply  the  result  by  the  index  of  the  given  power  for  a 
divisor. 

15.  Divide  the  dividend  by  the  divisor ^  and  consider  the  quotient 
the  next  figure  of  the  root. 

16.  Involve  the  whole  root  to  the  given  power,  and  subtract  it  from 
as  many  left  hand  periods  as  the  root  has  places  of  figures. 

17.  To  the  remainder  bring  down  the  next  period  for  a  new  divi- 
dend, to  which  find  a  new  divisor  as  before,  and  so  on  till  the  periods 
are  all  brought  down. 

18.  What  is  the  sursolid  or  5th  root  of  701,583,371,424  ? 

70  158337  1424(234 
2  "  =  3  2  subtrahend. 
2  *  X  5  =  divisor  8  0  )  3  8  1  dividend. 

2  3  «  =  6  43  6  3  43  subtrahend. 
23*X5  =  1399205)  5794907  dividend. 

234«  =  701583371424  subtrahend. 


19.  Observe  that  only  one  figure  is  brought  down  to  form  the  divi- 
dend, and  that  the  subtrahend  is  in  each  instance  taken  directly  from 
the  periods  in  the  top  line. 

20.  What  is  the  fifth  root  of  1,934,917,632 1  A.  72. 

21.  What  is  the  seventh  root  of  10,030,613,004,288  ^      A.  72. 

22.  What  is  the  tenth  root  of  3,486,784,401  ?  The  better  method 
IS  to  extract  the  5th  root  of  the  square  root.  A.  9. 

23.  If  the  amount  of  $100  for  8  years  at  compound  interest  be 
^159.38480745308416,  what  is  the  amount  for  the  first  year,  and 
what  is  the  rate  per  cent.]  A.  SI 06  ;  6  per  cent. 

Q.  What  is  the  fourth  or  biquadrate  root  of  256?— of  10,000?  In  the  rule 
which  is  applicable  to  all  powers,  what  is  the  direction  for  pointing  off?  11. 
What  is  the  rule  for  obtaining  the  dividend?  12,  13.  What,  for  finding  the 
divisor?  14.  What,  for  finding  the  second  figure  in  the  root?  16.  Describe 
the  rest  of  the  process  ?  16,  17 


248  ARITHMETIC. 

ALLIGATION. 

CII.  1.  Alligation  is  the  method  of  mixing  several  simples  of 
different  qualities,  so  that  the  compound  or  composition  may  be  of  a 
mean  or  middle  quality. 

2.  When  the  quantities  and  prices  of  the  several  things  or  simples 
are  given,  to  find  the  mean  price  or  mixture  compounded  of  them,  the 
process  is  called 

ALLIGATION     MEDIAL. 

3.  A  farmer  mixed  together  two  bushels  of  rye,  worth  50  cents  a 
bushel,  4  bushels  of  corn,  worth  60  cents  a  bushel,  and  4  bushels  of 
oats,  worth  30  cents  a  bushel ;  what  is  a  bushel  of  this  mixture  worth  1 

4.  In  this  example,  it  is 
2  bushels  at  $  .  5  0  cost  $1.00  Pl^in,  that  if  the  cost  of  the 

4 $.60 $2.40  whole  be  divided  by  the 

4 $.30 $1.20  whole  number  of  bushels, 

1~0 )  $4.60  (  4  6    the  quotient  will   be  the 

~^ —  ■     price  of  one  bushel  of  the 

mixture.       A-  46  cents. 
RULE. 

5.  Divide  the  whole  cost  by  the  whole  number  of  bushels,  <SfC  ;  the 
quotient  will  be  the  mean  price  or  cost  of  the  mixture. 

6.  A  grocer  mixed  10c wt.  of  sugar  at  $10  per  cwt.,  4  cwt.  at  $4 
per  cwt.,  and  8cwt.  at  7|  per  cwt.;  what  is  Icwt.  of  this  mixture 
worth  1 — what  is  5cwt.  worth  1 

A.  Icwt.  is  worth  $8,  and  5cwt.  is  worth  $40. 

7.  A  composition  was  made  of  51b.  of  tea  at  $1|  per  lb.,  91b.  at 
$1.80  per  lb.,  and  171b.  at  $1^  per  lb.;  what  is  a  pound  of  it  worth  1 

A.  $1.5463^+. 

8.  If  20  bushels  of  wheat,  at  $1.35  per  bushel,  be  mixed  with  15 
bushels  of  rye,  at  85  cents  per  bushel,  what  will  a  bushel  of  this  mix- 
ture be  worth?  A.  $l.l35j\+. 

9.  If  41b.  of  gold,  of  23  carats  fine,  be  melted  with  21b.  17  carats 
fine,  what  will  be  the  fineness  of  this  mixture  1  A.  21  carats. 

ALLIGATION    ALTERNATE. 

cm.  1.  Alligation  Alternate  is  the  process  of  finding  the 
proportional  quantity  of  each  simple,  from  having  the  mean  price  or 
rate,  and  the  mean  prices  or  rates  of  the  several  simples  given ;  con- 
sequently, it  is  the  reverse  of  alligation  medial,  and  may  be  proved 
by  it. 

2.  A  farmer  has  oats  worth  25  cents  a  bushel,  which  he  wishes  to 
mix  with  corn  worth  50  cents  per  bushel,  so  that  the  mixture  may  be 
worth  30  cents  per  bushel,  what  proportion  or  quantities  of  each 
must  he  take  1 

CII.     Q.  What  is  Alligation?  1.    Alligation  Medial ?  2.    Rule?  5. 
cm.  Q.  What  is  Alligation  Alternate  ?  1. 


30,1^-^-^0 


ALLIGATION.  249 

3.  In  this  example,  it  is  plain,  that  if  the  price  of  the  corn  had 
been  35  cents,  that  is,  had  it  exceeded  the  price  of  the  mixture  (30 
cents)  just  as  much  as  it  falls  short,  he  must  have  taken  equal  quan- 
tities of  each  sort ;  but,  since  the  difference  between  the  price  of  the 
corn  and  the  mixture  price  is  4  times  as  much  as  the  difference  be- 
tween the  price  of  the  oats  and  the  mixture  price,  4  times  as  much 
oats  as  corn  must  be  taken,  that  is,  4  to  1,  or  4  bushels  of  oats  to  1 
of  corn.  But  since  we  determine  this  proportion  by  the  differences, 
these  differences  will  represent  the  same  proportion. 

4.  These  are  20  and  5,  that  is,  20  bushels  of  oats  to  5  of  corn, 
which  are  the  quantities  or  proportions  required.  In  determining 
those  differences,  it  will  be  found  convenient  to  write  them  down  in 
the  following  manner : 

?1i"    ^!.?o^i'  ^'  ^^  ^^^^  ^^  recollected,  that  the 

oIH  -    5  C  ^^'    ^^^^^^^^^  between  50  and  30  is  20 ; 

1  that  is,  20  bushels  of  oats,  which  must 

stand  at  the  right  of  the  25,  the  price  of  the  oats,  or,  in  other  words, 
opposite  the  price,  that  is  connected  or  linked  with  the  50 ;  again, 
the  difference  between  25  and  30  =  5,  that  is,  5  bushels  of  corn  oppo- 
site the  50,  the  price  of  the  corn. 

6.  The  answer,  then,  is  20  bushels  of  oats  to  5  bushels  of  corn,  or 
in  that  proportion. 

7.  By  this  m'^de  of  operation,  it  will  be  perceived  that  there  is 
precisely  as  much  gained  by  one  quantity  as  there  is  lost  by  another, 
and  therefore  the  gain  or  loss  on  the  whole  is  equal. 

8.  The  same  will  be  true  of  any  two  ingredients  mixed  together  in 
the  same  way.  In  like  manner,  the  proportional  quantities  of  any 
number  of  simples  may  be  determined  ;  for,  if  a  less  be  linked  with 
a  greater  than  the  mean  price,  there  will  be  an  equal  balance  of  loss 
and  gain  between  every  two ;  consequently  an  equal  balance  on  tho 
whole. 

9.  It  is  obvious  that  this  principle  of  operation  will  allow  a  great 
variety  of  answers ;  for,  having  found  one  answer,  we  may  find  as 
many  more  as  we  please  by  only  multiplying  or  dividing  each  of  the 
quantities  found  by  2,  or  3,  or  4,  &c.;  for  if  two  quantities  of  two 
simples  make  a  balance  of  loss  and  gain,  as  it  respects  the  mean 
price,  so  will  also  the  double  or  treble,  the  ^  or  ^  part,  or  any  other 
ratio  of  these  quantities,  and  so  on  to  any  extent  whatever. 

10.  Proof. — We  will  now  ascertain  the  price  of  the  mixture  by 
the  last  rule,  thus : 

2  0  bushels  of  oats  at  25  cents  per  bushel =$  5.00 
_5  -  -  -  corn  at  50  -  -  -  -  =$2.50 
25 )7  .50(  30  cts.  A. 

RULE. 

11.  Having  reduced  the  several  prices  to  the  same  denomination^ 

Q.  Why  does  not  the  operation  affect  the  total  value  of  the  commodity?  7,  8, 
Why  is  not  the  result  confined  to  one  answer!  9.    Rule?  I  J,  12,  13. 


ihO  ARITHMETIC. 

connect  hj  a  line  each  price  that  is  less  than  the  mean  rate  with  one 
or  more  that  is  greater^  and  each  price  greater  than  the  mean  rate 
with  one  or  more  that  is  less. 

12.  Place  the  difference  bettveen  the  mean  rate  and  that  of  each  of 
the  simples  opposite  the  price  with  ivhich  they  are  connected. 

13.  Then,  if  only  one  difference  stands  against  any  price,  it  ex- 
presses the  quantity  of  that  price ;  but  if  there  be  several,  their  sum 
will  express  the  quantity. 

14.  A  merchant  has  several  sorts  of  tea,  some  at  10s.,  some  at 
lis.,  some  at  13s.  and  some  at  24s.  per  lb.;  what  proportions  of 
each  must  be  taken  to  make  a  composition  worth  12s.  per  lb.? 


15.  How  much  wine,  at  5s.  per  gallon  and  3s.  per  gallon,  must  be 
mixed  together,  that  the  compound  may  be  worth  4s.  per  gallon "? 

A.  1  gallon  of  each. 

16.  How  much  corn,  at  42  cents,  60  cents,  67  cents,  and  78  cents, 
per  bushel,  must  be  mixed  together,  that  the  compound  may  be  worth 
64  cents  per  bushel?  A.  14bu.  at  42c.;  3  at  60  ;  4  at  67 ;  22  at  78. 

17.  A  grocer  would  mix  different  quantities  of  sugar,  viz. — one  at 
20,  one  at  23,  and  one  at  26  cents  per  lb.;  what  quantity  of  each  sort 
must  be  taken  to  make  a  mixture  worth  22  cents  per  lb.? 

A.  51b.  at  20  cents ;  2  at  23  ;  2  at  26. 

18.  A  jeweller  wishes  to  procure  gold  of  20  carats  fine  from  gold 
of  16, 19, 21,  and  24  carats  fine  ;  what  quantity  of  each  must  he  take  ? 

A.  4,  1,  1,  4. 

19.  We  have  seen  that  we  can  take  3  times,  4  times,  ^,  j,  or  any 
proportion  of  each  quantity,  to  form  a  mixture. 

20.  Hence,  when  the  quantity  of  one  simple  is  given,  to  find  the 
proportional  quantities  of  any  compound  whatever,  after  having  found 
the  proportional  quantities  by  the  last  rule,  we  have  the  following 

RULE. 

21.  As  the  proportional  quantity  of  that  piece  whose  quantity  is 
given  is  to  each  proportional  quantity,  so  is  the  given  quantity  to  the 
quantities  or  proportions  of  the  compound  required. 

22.  A  grocer  wishes  to  mix  one  gallon  of  brandy,  worth  15s.  per 
gallon,  witfi  rum  worth  8s.,  so  that  the  mixture  may  be  worth  10s. 
per  gallon ;  how  much  rum  must  be  taken  ? 

23.  By  the  last  rule,  the  differences  are  5  to  2  ;  that  is,  the  pro- 
portions are  2  of  brandy  to  5  of  rum ;  hence,  he  must  take  2^  gallons 
of  rum  for  every  gallon  of  brandy.  A.  2\  gallons. 

24.  A  person  wishes  to  mix  10  bushels  of  wheat,  at  70  cents  per 
bushel,  with  rye  at  48  cents,  corn  at  36  cents,  and  barley  at  30  cents 
per  bushel,  so  that  a  bushel  of  this  mixture  may  be  worth  38  cents : 


ARITHMETICAL    PROGRESSION.  261 

what  quantity  of  each  must  be  taken  ]     We  find  by  the  last  rule,  that 
the  proportions  are  8,  2,  10,  and  32. 

Then,  as  8  :     2  :  :  1  0  :     2  i  bushels  of  rye.      \ 

8:10::10:124^  bushels  of  corn.    >  Answer. 

8:32::10:40''  bushels  of  barley .  ) 

25.  How  much  water  must  be  mixed  with  100  gallons  of  rum, 
worth  90cts.  per  gallon,  to  reduce  it  to  75cts.  per  gallon.  A.  20gal. 

26.  A  grocer  mixes  teas  at  $1.20,  $1,  and  60  cents,  with  201b.  at 
40c.  per  lb.;  how  much  of  each  sort  must  he  take  to  make  the  com- 
position worth  80c.  per  lb.       A.  20  at  $1.20,  10  at,  $1,  10  at  60c. 

27.  A  grocer  has  currants  at  4  cents,  6  cents,  9  cents,  and  11  cents 
per  lb.;  and  he  wishes  to  make  a  mixture  of  2401b.,  worth  8  cents  per 
lb.;  how  many  currants  of  each  kind  must  he  take  1  In  this  example 
we  can  find  the  proportional  quantities  by  linking,  as  before  ;  then  it 
is  plain  that  their  sum  will  be  in  the  same  proportion  to  any  part  ot 
their  sum,  as  the  whole  compound  is  to  any  part  of  the  compound, 
which  exactly  accords  with  the  principle  of  Fellowship. 

RULE. 

28.  As  the  sum  of  the  proportional  quantities  found  by  linking,  as 
before  :  is  to  each  proportional  quantity  :  :  so  is  the  whole  ^antity  or 
compound  required  :  to  the  required  quantity  of  each. 

We  will  now  apply  this  rule  in  performing  the  last  question. 

4 ^—3  flO:  3  ;  :240  :  721b.  at     4  cts.") 

—1    rru^^  J  10  :  1  ::  240  :  241b.  at     6  cts.  I  . 


®j     9 i     —2    ^^^"']  10  :  2:  :  240  :  481b.  at     9  cts.  f  ^• 

Li  1 1_4  Ll  0  :  4  :  :  2  4  0  :  9  6  lb.  at  1  1  cts.  J 

29.  A  grocer,  having  sugars  at  8c.,  12c.,  and  16c.  per  lb.,  v/ishes 
to  make  a  composition  of  1201b.,  worth  13c.  per  lb.;  what  quantity 
of  each  must  be  taken !  A.  301b.  at  8,  30lb.  at  12,  601b.  at  16. 

30.  How  much  water,  at  0  per  gal.,  must  be  mixed  with  wine,  at 
80c.  per  gal.,  so  as  to  fill  a  vessel  of  OOgal,  which  may  be  ofl*ered  at 
60c  per  gal.?        A.  bQ'l  gallons  of  wine,  and  33|  gallons  of  water. 

31.  How  much  gold,  of  15,  17,  18,  and  22  carats  fine,  must  be 
mixed  together,  to  form  a  composition  of  40  ounces  of  20  carats  fine  1 

A.  5oz.  of  15,  of  17,  of  18,  and  25oz.  of  22. 


ARITHMETICAL  PROGRESSION. 

CIV.  1.  Arithmetical  Progression,  or  Series,  is  any  rank  ot 
numbers  more  than  two,  that  increase  by  a  constant  addition,  or  de- 
crease by  a  constant  subtraction,  of  the  same  number. 

2.  The  Common  Difference  is  the  number  added  or  subtracted 
as  above.  .      ,    ij.  •        c 

3.  An  Ascending  Series  is  one  formed  by  a  continual  addition  ot 
the  common  difference,  as  2,  4,  6,  8,  10,  &c. 

CIV.  Q.  What  is  Arithmetical  Progression?  1.  What  the  Common  Differ 
ence?2.  An  Ascending  Series  ?  3.  A  Descending  Series  ?  4.  Give  an  63^ 
ample  of  each.  3, 4.     What  are  the  terms  ?  5. 


g§3  ARITHMETIC. 

4.  A  Descending  Arithmetical  Series,  is  one  formed  by  a  con- 
tinual subtraction  of  the  common  difference,  as  10,  8,  6,  4,  2,  &c. 

5.  The  Terms  are  those  numbers  that  form  the  series,  the  first  and 
last  of  which  are  called  the  Extremes,  and  the  other  the  Means. 

6.  In  Arithmetical  Progression  there  are  reckoned  five  terms,  any 
three  of  which  being  given,  the  remaining  two  may  be  found,  viz.— 

7.  1.  The  first  tefm;  2.  The  last  term;  3.  The  number  of  terms; 
4.  The  common  difference;  5.  The  sum  of  all  the  terms. 

8.  The  first  term,  the  last  term,  and  the  number  of  terms,  being 
given,  to  find  thcCommon  Difference  ; — 

9.  A  man  had  6  sons,  whose  several  ages  differed  alike:  the 
youngest  was  3  years  old,  and  the  oldest  28  ;  what  was  the  common 
difference  of  their  ages  ? 

10.  The  difference  between  the  youngest  son  and  the  eldest,  evi- 
dently shows  the  increase  of  the  3  years  by  all  the  subsequent  addi- 
tions, till  we  come  to  28  years ;  and,  as  the  number  of  these  additions 
are,  of  course,  1  less  than  the  number  of  sons  (5),  it  follows,  that,  if 
we  divide  the  whole  difference  (28-3= ),  25,  by  the  number  of  addi- 
tions (5),  we  shall  have  the  difference  between  the  ages  of  each,  that 
is,  the  common  difference.  Thus,  28 -  3=  25  ;  then,  25 ^ 5=  5  years, 
the  common  difference.  A.  5  years. 

11.  Hence,  to  find  the  common  difference, — Bivide  the  difference 
of  the  extremes  by  the  number  of  terms ,  less  1,  and  the  quotient  loill 
be  the  common  difference. 

12.  If  the  extremes  be  3  and  23,  and  the  number  of  terms  11,  what 
is  the  common  difference  1  A.  2. 

13.  A  man  is  to  travel  from  Boston  to  a  certain  place  in  6  days, 
and  to  go  only  5  miles  the  first  day,  increasing  the  distance  traveled 
each  day  by  an  equal  excess,  so  that  the  last  day's  journey  may  be 
45  miles  ;  what  is  the  daily  increase,  that  is,  the  common  difference  ? 

A.  8  miles. 

14.  If  the  amount  of  $1  for  20  years,  at  simple  interest,  be  $2.20, 
what  is  the  rate  per  cent.  1  In  this  example,  we  see  the  amount  of 
the  first  year  is  $1.06  and  the  last  year  $2.20,  consequently,  the  ex- 
tremes are  106  and  220,  and  the  number  of  terms  20. 

A.  $.06=6  percent. 

15.  A  man  bought  60  yards  of  cloth,  giving  5  cents  for  the  first 
yard,  7  for  the  second,  9  for  the  third,  and  so  on  to  the  last ;  what 
did  the  last  cost  1  Since,  in  the  last  example,  we  have  the  common 
difference  given,  it  will  be  easy  to  find  the  price  of  the  last  yard ;  for, 
as  there  are  as  many  additions  as  there  are  yards,  less  1,  that  is,  59 
additions  of  2  cents  to  be  made  to  the  first  yard,  it  follows,  that  the 
last  yard  will  cost  2x59=  118  cents  more  than  the  first,  and  the 
whole  cost  of  the  last,  reckoning  the  cost  of  the  first  yard,  will  be  118 
+5=$1.23.  ^.  S1.23. 

16.  Hence,  when  the  common  difference,  the  first  term,  and  the 

Q.  What  is  the  rule  for  finding  the  common  difference  ?  11.  For  finding  the 
iastterm?  16. 


ARITHMETICAL     PROGRESSION.  253 

number  of  terms,  are  given,  to  find  the  last  term. — Multiply  the  com- 
mon difference  by  the  number  of  tei-ms,  less  1,  and  add  the  first  term 
to  the  -product. 

17.  If  the  first  term  be  3,  the  common  difference  2,  and  the  number 
of  terms  11,  what  is  the  last  terra  ?  A.2Z. 

18.  A  man  went  from  Boston  to  a  certain  place  in  6  days,  travel- 
ing the  first  day  5  miles,  the  second  8  miles,  and  each  successive  day 
3  miles  farther  than  the  former ;  how  far  did  he  go  the  last  day ! 

A.  20  miles. 

19.  What  will  SI,  at  6  per  cent.,  amount  to,  in  20  years,  at  simple 
interest  '\  The  common  difference  is  the  6  per  cent.  ;  for  the  amount 
of  $1,  for  1  year,  is  $1.06,  and  1.06  +  8.06=81.12,  the  second  year, 
and  so  on.  A.  $2.20. 

20.  A  man  bought  10  yards  of  cloth,  in  arithmetical  progression  ; 
for  the  first  yard  he  gave  0  cents,  and  for  the  last  yard  he  gave  24 
cents ;  what  was  the  amount  of  the  whole  ]  In  this  example,  it  is 
plain  that  half  the  cost  of  the  first  and  last  yards  will  be  the  average 
price  of  the  whole  ;  thus,  6  cts.  +24  cts.==  30-^-2=  15  cts.,  average 
price  ;  then,  10  yds.  x  15=  $1.50,  whole  cost.  A.  $1.50. 

21.  Hence,  when  the  extremes,  and  the  number  of  terms,  are 
given,  to  find  the  sum  of  all  the  terms. — Multiply  half  the  sum  of  the 
extremes  by  the  number  of  terms,  and  the  product  will  be  the  answer. 

22.  If  the  extremes  be  3  and  273,  and  the  number  of  terms  40, 
what  is  the  sum  of  all  the  terms'?  A.  5520. 

23.  How  many  times  does  a  clock  strike  in  12  hours  ?      A.  78. 

24.  A  butcher  bought  100  oxen,  and  gave  for  the  first  ox  $1,  for  the 
second  $2,  for  the  third  $3,  and  so  on  to  the  last ;  how  much  did  they 
come  to  at  that  rate?  A.  85050. 

25.  What  is  the  sum  of  the  first  1000  numbers,  beginning  with 
their  natural  order,  1,  2,  3,  &c.  ]  A.  500500. 

26.  If  a  board,  18  feet  long,  be  2  feet  wide  at  one  end,  and  come 
to  a  point  at  the  other,  what  are  the  square  contents  of  the  board  1 

A.  18  feet. 

27.  If  a  piece  of  land,  60  rods  in  length,  be  20  rods  wide  at  one 
end,  and  at  the  other  terminate  in  an  angle  or  point,  what  number  of 
square  rods  does  it  contain  !  A.  600. 

28.  A  number  of  flat  stones  were  laid,  2  yards  distant,  for  the 
space  of  1  mile,  from  each  other,  and  the  first,  2  yards  from  a  certain 
basket.  How  far  will  that  man  travel  who  gathers  them  up  singly, 
and  returns  with  them  one  by  one  to  the  basket?       A.  881  miles. 

29.  A  person  traveling  into  the  country,  went  3  miles  the  first 
day,  and  increased  every  day's  travel  5  miles,  till  at  last  he  went  58 
miles  in  one  day  ;  how  many  days  did  he  travel  ? 

30.  We  found,  in  the  example  1,  the  difference  of  the  extreme  di- 
vided by  the  number  of  terms,  less  1,  gave  the  common  difference; 
consequently,  if,  in  this  example,  we  divided  (58—3=)  55,  the  differ- 
ence of  the  extremes,  by  the  common  difference,  5,  the  quotient  11, 

Q.  The  stun  of  all  the  terms  ?  21. 
22 


264  ARITHMETIC. 

will  be  the  number  of  terms,  less  1 ;  then,  1  +  11-12,  the  number  of 
terms.  ^-  12. 

31.  Hence,  when  the  extremes  and  common  difference  are  given, 
to  find  the  number  of  terms  : — Divide  the  difference  of  the  extremes 
by  the  common  difference,  and  the  quotient,  increased  by  1,  will  be  the 
ansioer. 

32.  If  the  extremes  be  3  and  45,  and  the  common  difference  6, 
what  is  the  number  of  terms  T  A.  S. 

33.  A  man  being  asked  how  many  children  he  had,  replied,  that 
the  youngest  was  4  years  old,  and  the  eldest  32,  the  increase  of  the 
family  having  been  1  in  every  4  years ;  how  many  had  he  1      ^.8. 


GEOMETRICAL  PROGRESSION. 

CV.  1.  Geometrical  Progression,  is  any  rank  or  series  of 
numbers,  which  increases  by  a  constant  multiplier,  or  decreases  by  a 
constant  divisor. 

2.  Thus,  3,  9,  27,  81,  &c.,  is  an  increasing  geometrical  series ;  and 
81,  27,  9,  3,  &c.,  is  a  decreasing  geometrical  series. 

3.  There  are  five  terms  in  Geometrical  Progression,  and,  like 
Arithmetical  Progression,  any  three  of  them  being  given,  the  other 
two  may  be  found,  viz  : — 

4.  1.  The  first  term.  2.  The  last  term.  3.  The  number  of  terms, 
4.  The  sum  of  all  the  terms.     5.  The  ratio. 

5.  A  man  purchased  a  flock  of  sheep,  consisting  of  9 ;  and  by 
agreement,  was  to  pay  what  the  last  sheep  came  to,  at  the  rate  of  $4 
for  the  first  sheep,  $12  for  the  second,  $36  for  the  third,  and  so  on, 
trebling  the  price  to  the  last ;  what  did  the  flock  cost  him  ? 

6.  We  may  perform  this  example  by  multiplication  ;  thus,  4x3x3 
X3x3x3x3x3x3= $26,244.  A.  But  this  process,  you  must  be  sen- 
sible, would  be,  in  many  cases,  a  very  tedious  one ;  let  us  see  if  we 
cannot  abridge  it  and  make  it  easier. 

7.  In  the  above  process,  we  discover  that  4  is  multiplied  by  3  eight 
times,  one  time  less  than  the  number  of  terms  ;  consequently,  the  8th 
power  of  the  ratio  3,  expressed  thus,  3"^,  multiplied  by  the  first  term, 
4,  will  produce  the  last  term.  But,  instead  of  raising  3  to  the  8th 
power  in  this  manner,  we  need  only  raise  it  to  the  4th  power,  then 
multiply  this  4th  power  into  itself;  for,  in  this  way,  we  do,  in  fact, 
use  the  3  eight  times,  raising  the  3  to  the  same  power  as  before  ; 
thus,  3*=81 ;  then  81  x  81 =6561 ;  this,  multiplied  by  4,  the  first  term, 
gives  $26,244,  the  same  result  as  before.  A.  $26,244. 

8.  Hence,  when  the  first  term,  ratio,  and  nuoiber  of  terms,  are 
given,  to  find  the  last  term. 

RULE. 

9.  Write  down  some  of  the  leading  powers  of  the  ratio,  with  the 

Q.  Number  of  terms  ?  31. 

CV.  Q.  What  is  Geometrical  Progression?  1.  What  are  the  terms?  4 
Give  examples  of  an  ascending  and  a  descending  series.  2. 


GEOMETRICAL    PROGRESSION.  255 

numbers  1,  2, 3,  cfc,  over  them,  being  their  several  indices.  Add  to- 
gether the  most  convenient  indices  to  make  an  index  less  by  1  than 
the  number  of  terms  sought. 

10.  Multiply  together  the  powers,  or  numbers  standing  under  those 
indices ;  and  their  product,  multiplied  by  the  first  term,  will  be  the 
term  sought. 

11.  If  the  first  term  of  a  geometrical  series  be  4,  and  the  ratio  3, 
what  is  the  1 1th  term  T 

1,  2,    3,    4,      5,  indices.  >     Note. — The  pupil  will  notice  that 

3,  9,27,81,243,  powers.  >  the    series  does  not  commence  with 

the  first    term,    but    with  the  ratio.      The    indices    5+3+2  =  10, 

and  the  powers  under  each,  243x27x9=59,049  ;  which,  multiplied 

by  the  first  term,  4,  makes  236,196,  the  11th  term  required. 

A.  236,196. 

12.  The  first  term  of  a  series,  having  10  terms,  is  4,  and  the  ratio 
3  ;  what  is  the  last  term  1  A.  78,732. 

13.  A  sum  of  money  is  to  be  divided  among  10  persons  ;  the  first  to 
have  810,  the  second  $30,  and  so  on,  in  threefold  proportion :  what 
will  the  last  have  ?  A.  $196,830. 

14.  A  boy  purchased  18  oranges,  on  condition  that  he  should  pay 
only  the  price  of  the  last,  reckoning  1  cent  for  the  first,  4  cents  for  the 
second,  16  cents  for  the  third,  and  in  that  proportion  for  the  whole  ; 
how  much  did  he  pay  for  theml  A.  $171,798,691.84. 

15.  What  is  the  last  term  of  a  series  having  18  terms,  the  first  of 
which  is  3,  and  the  ratio  3  ?  A.  $387,420,489. 

16.  A  butcher  meets  a  drover,  who  has  24  oxen.  The  butcher  in- 
quires the  price  of  them,  and  is  answered,  $60  per  head ;  he  immedi- 
ately offers  the  drover  $50  per  head,  and  would  take  all.  The  drover 
says  he  will  not  take  that ;  but,  if  he  will  give  him  what  the  last 
ox  would  come  to,  at  2  cents  for  the  first,  4  cents  for  the  second,  and 
so  on,  doubling  the  price  to  the  last,  he  may  have  the  whole.  What 
will  the  oxen  amount  to  at  that  rate?  A.  $167,772. 16. 

17.  A  man  was  to  travel  to  a  certain  place  in  4  days,  and  travel  at 
whatever  rate  he  pleased ;  the  first  day  he  went  2  miles,  the  second 
6  miles,  and  so  on  to  the  last,  in  a  threefold  ratio ;  how  far  did  he 
travel  the  last  day,  and  how  far  in  all  1 

18.  In  this  example,  we  may  find  the  last  term  as  before,  or  find  it 
by  adding  each  day's  travel  together,  commencing  with  the  first,  and 
proceeding  to  the  last,  thus  :  2  +  6+18  + 54 =80  miles,  the  whole  dis- 
tance traveled,  and  the  last  day's  journey  is  54  miles.  But  this  mode 
of  operation,  in  a  long  series,  you  must  be  sensible,  would  be  very 
troublesome.  Let  us  examine  the  nature  of  the  series,  and  try  to  in- 
vent some  shorter  method  of  arriving  at  the  same  result. 

19.  By  examining  the  series  2, 6,  18,  54,  we  perceive  that  the  last 
term  (54,)  less  2  (the  first  term,)=  52,  is  2  times  as  large  as  the  sura 
.ofthe  remaining  terms  ;  for  2  +  6  i- 18=26  ;  that  is,  54-2=52^2=  26; 
hence,  if  we  produce  another  term,  that  is,  multiply  54,  the  last  term, 

Q.  Give  the  rule  for  finding  the  last  term.  9,  10. 


256  .   ARITHMETIC. 

by  the  ratio  3,  making  162,  we  shall  find  the  same  true  of  this  also ; 
for  162-2  (the  first  term)  =:  160-^2  =  80,  which  we  at  first  found  to 
be  the  sum  of  the  four  remaining  terms;  thus,  2  +  6  +  18  +  54=^80. 
In  both  of  these  operations  it  is  curious  to  observe,  that  our  divisor, 
(2,)  each  time,  is  1  less  than  the  ratio  (3). 

20.  Hence,  when  the  extremes  and  ratio  are  given,  to  find  the 
sum  of  the  series,  we  have  the  following 

RULE. 

21.  Multiply  the  last  term  ly  the  ratio,  from  the  product  subtract 
the  first  term,  and  divide  the  remainder  by  the  ratio,  less  1 ;  the  quo- 
tient will  be  the  sum  of  the  series  required. 

22.  If  the  extremes  be  5  and  6,400,  and  the  ratio  6,  what  is  the 
whole  amount  of  the  series  1 


6400x0-5       ^^„„     , 
r.  _  I  =  7679,  Ans. 

23.  A  sum  of  money  is  to  be  divided  among  10  persons  in  such 
manner,  that  the  first  may  have  $10,  the  second  $30,  and  so  on,  in 
three-fold  proportion ;  what  will  the  last  have,  and  what  will  the 
whole  have  1 

24.  The  pupil  will  recollect  how  he  found  the  last  term  of  the  se- 
ries by  a  foregoing  rule  :  and,  in  all  the  cases  in  which  he  is  required 
to  find  the  sum  of  the  series,  when  the  last  term  is  not  given,  he  must 
first  find  it  by  that  rule,  and  then  work  for  the  sum  of  the  series  by 
the  present  rule.  A.  The  last,  $196,830 ;  and  the  whole,  $295,240. 

25.  A. hosier  sold  14  pair  of  stockings,  the  first  at  4  cents,  the 
second  at  12  cents,  and  so  on  in  geometrical  progression  ;  what  did 
the  last  pair  bring  him,  and  what  did  the  whole  bring  him"? 

A.  Last,  $63,772.92;  whole,  $95,059.36. 

26.  A  man  bought  a  horse,  and,  by  agreement,  was  to  give  a  cent 
for  the  first  nail,  three  for  the  second,  &c.;  there  were  four  shoes, 
and  in  each  shoe  eight  nails  ;  what  did  the  horse  come  to  at  that  rate "? 

A.  $9,205,100,944,259.20. 

27.  At  the  marriage  of  a  lady,  one  of  the  guests  made  her  a  pre- 
sent of  a  half-eagle,  saying  that  he  v/ould  double  it  on  the  first  day 
of  each  succeeding  month  throughout  the  year,  which,  he  said,  would 
amount  to  something  like  $100;  how  much  did  his  estimate  differ 
from  the  true  amount  ?  A.  $20,375. 

28.  If  our  pious  ancestors,  who  landed  at  Plymouth,  A.  D.  1620, 
being  101  in  number,  had  increased  so  as  to  double  their  number  in 
every  20  years,  how  great  would  have  been  their  population  at  the 
end  of  the  year  1840  ?  A.  206,747. 


ANNUITIES    AT    SIMPLE    INTEREST. 

CVI.     1.  An  Annuity  is  a  sum  of  money,  payable  every  year,  for 
a  certain  number  of  years,  or  forever. 
CVI.     Q.  What  is  an  aiiimity?  1. 


ANNUITIES    AT    SIMPLE    INTEREST.  267 

2.  When  the  annuity  is  not  paid  at  the  time  it  becomes  due,  it  is 
said  to  be  in  arrears. 

3.  The  sum  of  all  the  annuities,  such  as  rents,  pensions,  &c.  re- 
maining unpaid,  with  the  interest  on  each,  for  the  time  it  has  been 
due,  is  called  the  amount  of  the  annuity. 

4.  Hence,  to  find  the  amount  of  an  annuity — Calculate  the  interest 
on  each  annuity  for  the  time  it  has  remained  unpaid,  and  find  its 
amount ;  then  the  sum  of  all  these  several  amounts  will  be  the  amount 
required. 

5.  If  the  annual  rent  of  a  house,  which  is  S200,  remain  unpaid 
(that  is,  in  arrears)  8  years,  what  is  the  amount  ? 

6.  In  this  example,  the  rent  of  the  last  (8th)  year  being  paid  when 
due,  of  course  there  is  no  interest  to  be  calculated  on  that  year's  rent. 

The  amount  of  $200  for  7  years =S284 
The  amount  of  $200  for  6  years =$272 
The  amount  of  $200  for  5  years  =  $260 
The  amount  of  $200  for  4  years  =  $248 
The  amount  of  $200  for  3  years=$236 
The  amount  of  S200  for  2  years =$224 
The  amount  of  $200  for  1  year  =$212 
The  eighth  year,  paid  when  due, =$200 

$1,936  A. 


7.  If  a  man,  having  an  annual  pension  of  $60,  receive  no  part  of  it  till 
the  expiration  of  8  years,  what  is  the  amount  then  due  1  A.  $580.80. 

8.  What  would  an  annual  salary  of  $600  amount  to,  which  remains 
unpaid  (or  in  arrears)  for  2  years'? — 1,236.  For  3  years  1 — 1,908. 
For  4  years  1—2,616.  For  7  years  1—4,956.  For  8  years  1—5,808. 
For  10  years  1—7,620.  A.  Total,  $24,144. 

0.  What  is  the  present  worth  of  an  annuity  of  $600,  to  continue  4 
years  1  The  present  worth  [lxxxiii.]  is  such  a  sum  as,  if  put  at  in- 
terest, would  amount  to  the  given  annuity ;  hence, 

$600-^$l. 06=8566. 037,  present  worth,  1st  year. 
$600^$1.12  =  $535.714,  present  worth,  2d  year. 
$600 ^$1.18 =$508,474,  present  worth,  3d  year. 
$600 -^$1.24  =  $483.870,  present  worth,  4th  year. 
$2,094,095,  present  worth  required. 

10.  Hence,  to  find  the  present  worth  of  an  annuity. — Find  the  pre- 
sent worth  of  each  year  hj  itself,  discounting  from  the  time  it  becomes 
due,  and  the  sum  of  all  these  present  worths  ivill  be  the  answer. 

11.  What  sum  of  ready  money  is  equivalent  to  an  annuity  of  $200, 
to  continue  3  years,  at  4  per  cent.1  A.  $556,063. 

12.  What  is  the  present  worth  of  an  annual  salary  of  $800,  to 
continue  2  years  1—1,469.001.  3  years  1—2,146.9671  6  years  1— 
3,407.512.  A.  Total,  $7,023.48. 

Q.  What  is  meant  by  arrears  and  amount  ?  2, 3.  Rule  for  finding  the  amount  ?  4. 
For  finding  the  present  worth?  10. 
22* 


268 


ARITHMETIC. 


ANNUITIES  AT  COMPOUND  INTEREST 

CVII.     1.   The  amount  of  an  annuity  at  simple  and  compound  iu' 
ierest  is  the  same,  excepting  the  difference  in  interest. 

2.  Hence,  to  find  the  amount  of  an  annuity  at  compound  interest. — 
Proceed  as  i?i  cvi.,  reckoning  compound  instead  of  simple  interest. 

3.  What  will  a  salary  of  $200  amount  to,  which  has  remained  un  • 
paid  for  3  years'? 

The  amount  of  $200  for  2  years  =  $224.72 
The  amount  of  $200  for  1  year  =$212.00 

\  The  3d  year =$200.00 

yl.  $636.72 


4.  If  the  annual  rent  of  a  house,  which  is  $150,  remain  in  arrears 
for  3  years,  what  will  be  the  amount  due  for  that  timxC  ]  A.  477.54. 

5.  Calculating  the  amount  of  the  annuities  in  this  manner,  for  a 
long  period  of  years,  would  be  tedious.     This  trouble  will  be  pre 
vented,  by  finding  the  amount  of  $1,  or  .£1,  annuity,  at  compound  in 
terest,  for  a  number  of  years,  as  in  the  following 

TABLE    I. 

Showing  the  amount  of  $1,  or  £\,  annuity,  at  6  j)er  cent,,  compound  interest,  for  any 

number  of  years,  from  1  to  50. 


1 

6  per  cent. 
I.OIOO 

Y. 

Ti 

12 
13 
14 

fs 

16 
17 

18 
19 
20 

6  per  cent. 

Y. 
21 

6  per  cent. 

Y. 

6  per  cent.  | 

Y. 

41 
42 
43 
44 
4-5 
46 
47 
48 
49 
50 

6  per  cent. 

14.9716 
16.8699 

18.882'! 

39.9927 

31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

84.8010! 
90.8897 
97.3431 

lt;5  0407 
175.9495 
187.5064 
199.7568 
212.7423 

2 
3 

2.0600 
3.1836 

22 
23 
24 
25 

26 

43.3922 
46.9958 

4 
5 

6 
7 
8 
9 
10 

4.3746 

21.0150 

50.8155 
54.8045 
59.1563 

104.1837 
i  11.4347 

5.6371 
6.9753 
8.3938 

23.2759 
25.6725 

119.1208 
127.2681 
135.9042 
145.0584 
154.7619 

226.5068 
231.0972 
245.9630 
261.7208 
278.4241 

28.2123 
30.9056 
33.7599 
3617855 

2763.7057 

ir  68.5281 

9.8974 

11.4913 
13.1807 

29 
310 

73.6397 

79.0.-S-1 

It  is  evident,  that  the  amount  of  $2  annuity  is  2  times  as  much  aa 
one  of  $1 ;  and  one  of  $3,  3  times  as  much. 

6.  Hence,  to  find  the  amount  of  an  annuity,  at  0  per  cent. — Find, 
by  the  table,  the  amount  0/  $1,  at  the  given  rate  and  time,  and  multi- 
ply it  by  the  given  annuity,  and  the  product  will  be  the  amount  re- 
quired. 

7.  What  is  the  amount  of  an  annuity  of  $120,  which  has  remained 
unpaid  15  years'?  The  amount  of  $1,  by  the  table,  we  find  to  be 
$23.2759  ;  therefore,  $23.2759  x  120=$2,793. 108.  A. 

8.  What  will  be  the  amount  of  an  annual  salary  of  $400,  which 
has  been  in  arrears  2  years  1—824.  3  years]— 1,273.44.  4  years?— 

CVII.  Q.  What  is  meant  by  the  amount  of  an  annuity  at  compound  interest?  1. 
How  is  it  found?  2. 


ANNUITIKS    AT    COMl'Ol'xVl)    iNTtllEST. 


259 


1,749.84.    6  years]— 2,790.12.    12  years  1—6,747.96.    20  years  1— 
14,714.2.  A.  Total,  S28,099.56. 

9.  If  you  lay  up  $100  a  year,  from  the  time  you  are  21  years  of  age 
till  you  are  70,  what  will  be  the  amount  at  compound  interest  \ 

A.  $26,172.08. 

10.  What  is  the  present  worth  of  an  annual  pension  of  $120,  which 
is  CO  continue  3  years  1 

1 1.  In  this  example,  the  present  worth  is  evidently  that  sum  which, 
at  compound  interest,  would  amount  to  as  much  as  the  amount  of  the 
given  annuity  for  the  three  years.  Finding  the  amount  of  $120  by 
the  table,  as  before,  we  have  $382,082  ;  then,  if  we  divide  $382,032 
by  the  amount  of  $1,  compound  interest,  for  3  years,  the  quotient 
will  be  the  present  worth.  This  is  evident  from  the  fact,  that  the 
quotient^  multiplied  by  the  amount  of  $1,  will  give  the  amount  of 
$120,  or,  in  other  words,  $382,032.  The  amount  of  $1  for  3  years 
at  compound  interest  is  $1.10101;  then,  $382. 032^$1. 19101  = 
$320,763,  A. 

12.  Hence,  to  find  the  present  worth  of  an  annuity. — Find  Us 
amount  in  arrears  for  the  ivhole  time ;  this  amount,  divided  hj  the 
amount  of  $1  for  said  time,  ivill  be  the  present  loorth  required. 

13.  Note. — The  amount  of  SI  may  be  found,  ready  calculated,  in 
the  table  of  compound  interest,  [lxxxii.] 

14.  What  is  the  present  worth  of  an  annual  rent  of  $200,  to  con- 
tinue 5  years  ]  A.  $842,472. 

15.  The  operations  in  this  rule  may  be  much  shortened  by  calcu- 
lating the  present  worth  of  $1  for  a  number  of  years,  as  in  the  fol- 
lowing 

TABLE    II. 

Showiiig  tlie  present  worth  of  $1,  or  £\,  annuity,  at  6  per  cent.,  compound  interest,  for 

any  number  of  years,  from  I  to  32. 


Y. 

1 
2 

1 

4 

5 

\6 

7 

fi  per  cent. 

Y. 

9 
10 
11 
l2 
13 
14 
15 
16 

ti  per  cent. 

Y. 

ri  per  cent.  1 

Y. 

25 

1 

28 
29 

6  per  cent. 

12.78335 

0.94339 
1.83339 
2.67301 
3.46510 
4.21236 

6.80169 
"J7. 36008 

7.88'687 

17il0.47726 

18:10.82760 

13.00316 

19 
20 
21 

22 
23 
'24 

11.15811 
11.46992 

13.21053 
13.40616 

8.38384 
8.85268 
9.29498 

11.76407 
12.04158 
12.30338 

13.59072 

4.91732 

5.58238 

30 
31 
32 

13.76483 
13.92908 
14.08398 

9.71225 

8 

1  6.20979 

10.10589 

12.55035 

16.  To  find  the  present  worth  of  any  annuity  by  this  table,  we 
have  only  to  multiply  the  present  worth  of  $1,  found  in  the  table,  by 
the  given  annuity,  and  the  product  will  be  the  present  worth  required. 

17.  What  sum  of  ready  money  will  purchase  an  annuity  of  8300, 
to  continue  10  years'?  The  present  worth  of  $1  annuity,  by  the  Ta- 
ble, for  10  years,  is  $7.36008  ;  then  7.36008  x300=$2,208.024.  A. 

Q.  How  is  the  present  worth  found?  12. 


260  AUITHMETIC. 

18.  What  is  the  present  worth  of  a  yearly  pension  of  $60,  to  con 
tinue  2  years?— 110.0034.  3  years  1—160.3806.  4  years  1—207.906 
8  years  ?— 372.5874.  20  years  1—688. 1952.  30  years  1—825.8898 
Total,  $2,364.9624. 

19.  What  salary,  to  contimie  10  years,  will  $2,208,024  purchase  1 
This  exampleisthe  17th  example  reversed;  consequently,  $2,208,024 
-^7.36008  =  300,  the  annuity  required.  A.  $300. 

20.  Hence,  to  find  that  annuity  which  any  given  sum  will  pur- 
chase.— Divide  the  given  sum  by  the  present  worth  of  ^l  annuity  for 
the  given  time,  found  by  Table  n.;  the  quotient  will  be  the  annuity  re- 
quired. 

21.  What  salary,  to  continue  20  years,  will  $688,195  purchase  1 

A.  $00+. 

22.  What  annuity,  to  continue  10  years,  is  equivalent  to  $3,680.04  '* 

A.  $500. 

23.  To  divide  any  sum  of  money  into  annual  payments,  which, 
when  due,  shall  form  an  equal  amount  at  compound  interest. — First 
find  an  equivalent  annuity  as  above,  (20,)  then  its  present  worth  for 
each  required  period  of  time.  (15.) 

24.  A  certain  manufacturing  establishment  in  Massachusetts  was 
actually  sold  for  $27,000,  and  the  sum  divided  into  four  notes,  pay- 
able annually,  so  that  the  principal  and  interest  of  each,  when  due, 
should  form  an  equal  amount,  at  compound  interest,  and  the  several 
principals,  when  added  together,  should  make  $27,000 ;  now,  what 
were  the  principals  of  said  notes  1* 

The  first  note  is  $7,350,915;  amount  for  1  year,  $7,791.97032. 
The  second  note  is  $0,934.825 ;  amount  for  2  years,  $7,791 .97032. 
The  third  note  is  $6,542.288 ;  amount  for  3  years,  $7,791.97032. 
The  fourth  note  is  $6,171.970;  amount  for  4  years,  $7,791 .97032. 
Proof— $27,000,  lacking  2  mills. 


PERMUTATION. 

CVIII.  1.  Permutation  is  the  method  of  finding  how  many  dif- 
ferent ways  any  number  of  things  may  be  changed. 

2.  How  many  changes  may  be  made  of  the  first  three  letters  of  the 
alphabet  1  In  this  example,  had  there  been  but  two  letters,  they 
could  only  be  changed  twice  ;  that  is,  a,  b,  and  b,  a ;  that  is,  1  x  2=  2 ; 
but,  as  there  are  three  letters,  they  may  be  changed  1  x  2  x  3=  G 
times,  as  follows — 1,  a,  b,  c  ;  2,  a,  c,  b ;  3,  b,  a,  c  ;  4,  b,  c,  a ;  5,  c, 
b,  a;  6,  c,  a,  b. 

Q.  What  is  the  rule  for  finding  what  sum  a  given  annuity  will  purchase  ?  20. 

CVIII.     Q.   What  is  Permutation?  1.     How  many  changes  can  be  made 

with  the  first  three  letters  of  the  alphabet?  2.     What  are  they?  2.     Rule?  3. 

*The  annuity  which  $27,000  will  purchase,  found  as  before,  is  7,791.97032  +  .  To 
obtain  an  exact  result,  we  must  reckon  the  decimals,  which  were  rejected  in  forming  the 
tables.    This  makes  the  last  divisor  3.4651(156. 


POSITION.  261 

3.  Hence,  to  fmd  the  number  of  different  changes  or  permutations 
which  may  be  made  with  any  given  number  of  different  tilings. — 
Multiply  together  all  the  terms  of  the  natural  series,  from  1  up  to  the 
given  number,  and  the  last  product  will  he  the  number  of  changes  re- 
quired. 

4.  Plow  many  different  ways  may  the  first  five  letters  of  the  alpha- 
bet be  arranged  1  A.  120. 

5.  How  many  changes  may  be  rung  on  15  bells,  and  in  what  time 
may  they  be  rung,  allowing  3  seconds  to  every  round  T 

A.  1,307,074,308,000  changes;  3,923,023,104,000  seconds. 

6.  What  time  will  it  require  for  10  boarders  to  scat  themselves 
differently  every  day  at  dinner,  allov/ing  365  days  to  the  year  ? 

A.  9,941  ^if  years. 

7.  Of  how  many  variations  will  the  2(5  letters  of  the  alphabet  ad- 
mit? A.  403,291,461,126,005,635,534,000,000. 


POSITION. 

Position  is  a  rule  which  teaches,  by  the  use  of  supposed  numbers, 
to  find  true  ones.  It  is  divided  into  two  parts,  called  Single  and 
Double. 

SINGLE  POSITION. 

CIX.  1.  Single  Position  teaches  to  resolve  those  questions 
whose  results  are  proportional  to  their  suppositions. 

2.  A  schoolmaster,  being  asked  hov.r  many  scholars  he  had,  replied, 
"  If  I  had  as  many  more  as  I  now  have,  one  half  as  many  more,  one 
third  and  one  fourth  as  many  more,  I  should  have  296."  How  many 
had  he  I 

We  have  now  found  that  we  did  not  suppose 
the  right  number.  If  we  had,  the  amount  would 
have  been  296.  But  24  has  been  increased  in 
the  same  manner  to  amount  to  74,  that  some 
unknown  number,  the  true  number  of  scholars, 
must  be,  to  amount  to  296.  Consequently,  it  is 
obvious,  that  74  has  the  same  ratio  to  290  that 
24  has  to  the  true  number.  The  question  may, 
therefore,  be  solved  by  the  follovv  ing  statement : 
As  74  :  296  :  :  24  :  96, /i. 

3.  This  answer  we  prove  to  be  right  by  increasing  it  by  itself,  one 
half  of  itself,  one  third  of  itself,  and  one  fourth  of  itself,  as,  96  +  96  + 
48  +  32+24=296. 

RULE. 

4.  Suppose  any  number  you  choose,  and  proceed  with  it  in  the  same 
manner  you  loould  icith  the  answer,  to  see  if  it  were  right ;  then  say^ 
as  this  result :  the  rcsidt  in  the  question  :  :  the  supposed  number  : 
number  sought. 

I. IX.     Q.  What  is  Position?     Sin-lc  Position?  1.     Rule?  4. 


Suppose  he  had  24 

As  many  more=  24 

\  as  many=  12 

5  as  many=   8 

I  as  many=  _6 

74 


262 


ARITHMETIC. 


5.  James  lent  William  a  sum  of  money  on  interest,  and  in  10  years 
it  amounted  to  $1,600 ;  what  was  the  sum  lent  ?  A.  $1,000. 

G.  Three  merchants  gained,  by  trading,  $1,920,  of  which  A  took  a 
certain  sum,  B  took  3  times  as  much  as  A,  and  C  four  times  as  much 
as  B  ;  what  share  of  the  gain  had  each  1 

A.  A,  $120;  B,  $360;  C,  $1,440. 

7.  A  person  having  about  him  a  certain  number  of  crowns,  said, 
if  a  third,  a  fourth,  and  a  sixth,  of  them  were  added  together,  the 
sum  would  be  45  ;  how  many  crowns  had  he  ]  A.  60. 

8.  What  is  the  age  of  a  person,  who  says,  that  if  yV  of  the  years 
he-^  has  lived  be  multiplied  by  7,  and  f  of  them  be  added  to  the  pro- 
duct, the  sum  would  be  292 1  A.  60  years. 

9.  What  number  is  that,  which,  being  multiplied  by  7,  and  the  pro- 
duct divided  by  6,  the  quotient  will  be  14  ]  A.  12. 

DOUBLE  POSITION, 
ex.     1.  Double  Position  teaches  to  solve  questions  by  means  of 
two  supposed  numbers. 

2.  In  Single  Position,  the  number  sought  is  always  multiplied  or 
divided  by  some  proposed  number,  or  increased  or  diminished  by  itself, 
or  some  known  part  of  itself,  a  certain  number  of  times.  Consequently, 
the  result  will  be  proportional  to  its  supposition,  and  but  one  supposi- 
tion will  be  necessary  ;  but,  in  Double  Position,  we  employ  two,  for 
the  results  are  not  proportional  to  the  suppositions. 

3.  A  gentleman  gave  his  three  sons  $10,000,  in  the  following  man- 
ner ;  to  the  second  $1000  more  than  to  the  first,  and  to  the  third  as 
many  as  to  the  first  and  second  1     What  was  each  son's  part  1 


Let  us  suppose  the  share  of  the  first  1,000"' 
Then  the  second=2,000 
Third  ^3,000 


The  shares  of  all  the 
sons  will,  if  our  suppo- 
sition be  correct,  am't 
to  10,000;  but,  as  they 
Total,  6,000     amount  to  $6,000  only 
This  subtracted  from  10,000,  leaves  4,000  J  we  call  the  error  4000. 
Suppose  again,  that  the  share  of  the  first  was  1,500~1 
Then  the  second =2,500 

Third  =  4,000        We  perceive 
>  the  error  in  this 


8,000 


case  to  be  2000. 


2,000^ 

4.  The  first  error,  then,  is  $4,000,  and  the  second  $2,000.  Now, 
the  difference  between  these  errors  would  seem  to  have  the  same  re- 
lation to  the  difference  of  the  suppositions,  as  either  of  the  errors 
would  have  to  the  difference  between  the  supposition  which  produced 
it,  and  the  true  number.  We  can  easily  make  this  statement,  and 
ascertain  w^hether  it  will  produce  such  a  result : 

5.  As  the  difference  of  errors,  2,000  ;  500  difference  of  supposi- 
tions ::  either  of  the  errors  (say  the  first,)  4,000  : 1,000,  the  difference 


MENSURATION.  263 

between  its  supposition  and  the  true  number.  Adding  this  diftcrence 
to  1,000,  the  supposition,  the  amount  is  2,000  for  the  share  of  the  first 
son  :  then  $3,000  that  of  the  second,  $5,000  that  of  the  third,  Ans. 
For  2,000 +  3,000 +  5,000  =-10,000,  the  whole  estate. 

6.  Had  the  supposition  proved  too  great,  instead  of  too  small,  it  is 
manifest  that  we  must  have  subtracted  this  diflerence.  The  differ- 
ences between  the  results  and  the  result  in  the  question  are  called  er- 
rors ;  these  are  said  to  be  alike,  when  both  are  either  too  great  or  toa 
small ;  unlike,  when  one  is  too  great,  and  the  other  too  small. 

RULE. 

7.  Suppose  any  tivo  numbers,  and  proceed  loilh  each  according  to 
the  manner  described  in  the  question,  and  see  how  much  the  result  of 
each  differs  from  that  in  the  question. 

8.  Then  say,  as  the  difference*  of  the  errors  :  the  difference  of  the 
suppositions ::  either  error :  difference  between  its  supposition  and  the 
number  sought. 

9.  Three  persons  disputing  about  their  ages,  says  B,  "  I  am  10 
years  older  than  A  ;"  says  C,  "  I  am  as  old  as  you  both  :"  now,  what 
were  their  several  ages,  the  sum  of  them  all  being  100  "? 

A.  A's,  20;  B's,  30;  C's,  50. 

10.  Two  persons,  A  and  B,  have  the  same  income  :  A  saves  \  of 
his  yearly  ;  but  B,  by  spending  $150  per  annum  more  than  A,  at  the 
end  of  8  years,  finds  himself  $400  in  debt ;  what  is  their  income,  and 
what  does  each  spend  per  annum  ! 

A.  A's  income  $400  ;  A  spends  $300 ;  B  $450. 

11.  There  is  a  fish  whose  head  is  8  feet  long,  his  tail  is  as  long  as 
his  head  and  half  his  body,  and  his  body  is  as  long  as  his  head  and 
tail;  what  is  the  whole  length  of  the  fish.  A.  64  feet. 

12.  A  laborer  contracted  to  work  80  days  for  75  cents  per  day,  and 
to  forfeit  50  cents  for  every  day  he  should  be  idle  during  that  time. 
He  received  $25  :  now  how  many  days  did  he  work,  and  how  many 
days  was  he  idle  1  A.  52  days ;  idle  28. 


MENSURATION. 

CXI.     1.  Mensuration  is  the  measuring  of  Surfaces  and  Solids. 

OF      ANGLES. 

2.  An  Angle  is  the  inclination  or  opening  of  two  Imes  that  meet 
each  other,  as  in  the  Figures  on  next  page.  The  point  of  intersec- 
tion is  called  the  Angular  point ;  and  in  common  language,  the  Angle. 

3.  An  Angle  is  greater  or  less,  not  according  to  the  length  of  the 

CXI.  Q.  What  is  Mensuration  ?  1.  An  Angle?  2.  The  point  of  intersec- 
tion ?  2.     How  is  the  size  of  an  angle  determined?  3. 

*  The  difference  of  the  errora,  when  alike,  will  be  one  subtracted  from  the  other 
when  unlike,  one  added  to  the  other. 


264 


ARITHMETIC. 


lines,  but  according  as  tliey  are  more  or  less  inclined  or  opened  ;  thus, 
the  angle  at  C,  below,  is  the  greatest  of  the  three. 

Fia.  I  Fio.  2.  Fio.  3. 


Right  Angle,     A  Acute  Angle.  B  Obtuse  Angle,  C 

4.  A  Right  Angle  is  one  formed  by  a  line  drawn  perpendicular  to 
another  :  as  A,  in  Fig.  1. 

5.  Oblique  Angles  are  those  formed  by  oblique  lines,  and  are 
either  Acute  or  Obtuse  ;  as  B  and  C. 

6.  An  Obtuse  Angle  is  greater,  and  an  Acute  Angle  is  less  than 
a  right  angle. 

OF     TRIANGLES. 

7.  A  Triangle  is  a  plane*  figure  that  has  three  sides  «.nd  three 
angles  ;  as  in  the  following  Figures. 

Fio.  4.  Fig.  5.  Fig.  6.  Fig.  7. 


4.J 


Equilateral.  Isosceles.  Scalene.  Right. 

8.  An  Equilateral  Triangle  has  three  equal  sides.  [Fig. 

9.  An  Isosceles  Triangle  has  two  equal  sides.  [Fig.  5. J 

10.  A  Scalene  Triangle  has  three  unequal  sides.   [Fig.  6.] 

11.  A  Right- Angled  Triangle  has  one  right  angle.  [Fig.  7.] 

12.  An  Obtuse-Angled  Triangle  has  an  obtuse  angle.  [Fig.  6.] 

13.  An  Acute-Angled  Triangle  has  three  acute  angles.  [Fig.  4.] 

14.  In  a  right-angled  triangle  the  longest  side  is  the  Hypothenuse, 
and  the  other  two  sides  the  Legs,  or  the  Base 
and  Perpendicular.     In   other  triangles  the 
longest  side  is  usually  considered  the  Base. 

15.  In  every  right-angled  triangle, — The 
square  of  the  hypothenuse  is  equal  to  the  sum 
of  the  squares  of  the  other  two  sides ;  as,  bO^  — 
40^  +  30^     [Fig.  8.] 

16.  Hence,  to  find  the  different  sides,  we  may  proceed  as  follows  : 
To  find  the  hypothenuse. — Add  the  squares  of  the  two  legs  to- 
gether, and  extract  the  square  root  of  that  sum.     To  find  either  leg. 
From  the  square  of  the  hypothenuse  subtract  the  square  of  the  given 
leg,  and  the  square  root  of  the  remainder  ivill  he  the  other  leg. 


.  How  is  a  right  angle  formed  ?  4. 

Acute?  6.     What  is  a  triangle  ?  7.     An  equilateral  triangle?  8.     An  Isos- 


What  are  oblique  angles  ?  5.     Obtuse 
An  equilateral  triangle  ?  8. 
celes?  9.     Scalene?  10.     Right  angled  triangle  ?  11.     Obtuse  angled  triangle? 

tiames  of  the  sides  in  a  right 
On  what  principle  is  each 


ngie 
12.     Acute  angled  triangle  ?  ]  3.     What  are  the  names  of  the  sides  in  a  right 
angled  triangle ?  14.     How  are  each  found?  16. 
operation  based  ?  15. 

*  Plane,  [L.  Planus.]  An  even  or  level  surtace,  like  plain  in  commoa  language.    An 
instrument  used  in  smoothing  boards. 


MLXSURATION.  ^  265 

Try  The  learner  will  discover  the  application  of  the  rule  better  by- 
drawing  a  triangle  on  his  slate,  like  Fig.  8,  and  noting  the  sides  which 
are  intended  to  correspond  with  those  which  are  given  in  the  ques- 
tion. Indeed,  without  some  such  illustration,  he  will  scarcely  be 
able  to  apply  the  rule  at  all,  except  in  cases  where  the  particular 
sides  are  designated. 

17.  Required  the  hypothenuse  of  a  right-angled  triangle  whose 
legs  are  24  and  32  feet.  A.  40  feet. 

18.  Required  the  base  of  a  right-angled  triangle  the  other  sides  of 
which  are»respectively  15  and  25  feet.  A.  20  rods. 

19.  Suppose  a  lot  of  land  lies  in  the  form  of  a  right-angled  triangle, 
and  that  the  longest  side  is  100  rods  and  the  shortest  00  ;  what  is  the 
distance  around  if?  A.  240  rods. 

20.  A  river  80  yards  wide  passes  by  a  fort,  the  walls  of  which  are 
face  of  a"  Rhombus"  ul^  k.tbfi.distance  /^Qm  tbe.ton.nfjbe  wall  to 
or  parallelogram  of  the  same  length,  but  whose  breadth  is  its  perpen- 
dicular height. 

39.  A  Diagonal  is  a  line  that  passes  across  a  quadrilateral  from 
one  angle  to  its  opposite  one. 

40.  The  diagonal  of  every  parallelogram  divides  it  into  two  equal 
parts,  as  in  Fig.  16. 

4.1.  Hence  the  area  of  every  right-angled  triangle  is  just  half  as 
straight  line  to  the  water  is  120  feet,  and  the  distance  nom  the  lOOt 
89ft. ;  what  is  the  height  of  the  tree  1  (89  +  1  base.)     A.  79.372ft.+  . 

23.  When  two  ships,  which  sailed  from  the  same  port,  have  gone, 
one  due  north  40  leagues,  and  the  other  due  east  30  leagues,  how  far 
are  they  apart  then  ]  ^ .  50  leagues. 

Fio.  9. 

24.  There  are  three  towers,  A,  B,  and  C,  ^  _..— --nfi 
standing  in  a  direct  line,  the  heights  of  which  '  '  ^^ 
are  64,  90,  and  50  feet  respectively.  The 
distance  between  the  top  of  the  tower  A  and 
that  of  B,  is  97  feet,  and  the  distance  between 
the  bottom  of  the  tower  B  and  that  of  C  is  76 
feet.  From  these  data  please  inform  me  what  are  the  several  dis- 
tances from  the  top  of  A  to  the  bottom  of  B,  from  the  top  of  B  to  the 
bottom  of  A,  from  the  bottom  of  A  to  the  bottom  of  B,  from  the  bot- 
tom of  B  to  the  top  of  C,  from  the  bottom  of  C  to  the  top  of  B,  and 
from  the  top  ofB  to  the  top  of  C.  A.  D  E,  93.45  +  ;  A  E,  113.26  +  ; 
B  D,  129.74+  :  C  E,  90.97+  ;  B  F.  117.79+  ;  B  C,  85.883+. 

1,  42.) 

48.  Or,  halve  the  sum  of  the  three  sides,  subtract  the  three  sides 

Q.  With  what  is  the  area  of  a  rhomboid  or  rhombus  compared  ?  38.  What  is 
diagonal  ?  39.  What  comparison  is  made  between  the  area  of  a  right-angled 
iangle  and  that  of  a  square?  41.  What  two  equal  divisions  may  be  made  of 
:  1  obhque-angled  triangle?  42.  What  is  the  inference  ?  43.  What  is  the  rule 
t  finding  the  area  of  a  square  ?  44. — of  a  rhomboid  ?  45. — of  a  right-angled  tri- 
igle  ?  46. — of  an  oblique-angled  triangle  ?  47,  48. 


E     It 


•  ^ 


266 


ARITHMETIC. 


26.  A  gentleman  has  a  garden  in  the  form  of  an  equilateral  tri- 
angle, the  sides  of  which  are  each  50  feet ;  at  each  corner  of  the  gar- 
den stands  a  tower ;  the  height  of  the  tower  A  is  30  feet,  that  of  B 
34  feet,  and  that  of  C  28  feet.  At  what  distance  from  the  bottom  of 
each  of  these  towers  must  a  ladder  of  the  same  length  with  each  side 
be  placed,  that  it  may  just  reach  the  top  of  each  tower,  allowing  the 
ground  of  the  garden  to  be  horizontal  ? 

A.  40ft. ;  36.G6ft.  +  ;  41.42ft. 

OF      SURFACES. 

27.  A  Quadrilateral  has  four  sides  and  four  angles.  Paral- 
lelogram is  a  general  name  for  all  quadrilateral  figures,  that  have 
at  least  their  opposite  sides  and  angles  equal ;  as  below. 


Fig.  10. 


Fig.  11. 


Fig.  12. 


Fig.  13. 


Equilateral.  Isosceles.  Scalene.  Right. 

8.  An  Equilateral  Triangle  has  three  equal  sides.  [Fig.  4.] 

9.  An  Isosceles  TRiANfiT.p^  has  t.vvnonnoic-;ric.=    rT?;-  '^„V;„  in 
ttiigies  equal,  two  ot  its  angles  bemg  acute,  and  two  obtuse,  as  Jb  ig.  12. 

31.  A  RnoMBUS  is  a  quadrilateral  that  has  its  sides  equal,  but  its 
angles  like  those  of  a  rhomboid,  as  Fig.  13. 

32.  A  Polygon  is  a  rectilineal  figure  of  more  than  four  sides, 
which  when  they  are  all  equal,  form  regular  polygons.  Squares  are 
also  regular  figures ;  so  are  Triangles,  when  they  are  equilateral. 

33.  The  Perimeter  of  any  plane  rectilineal*  figure,  is  the  entire 
distance  round  it ;  and  is  found  by  adding  together  all  the  sides  that 


Fig.  14. 


bound  it. 

34.  A  Circle  is  a  plane  figure  bounded  by 
a  curved  line,  called  the  Circumference  or 
Periphery  ;  which  is  every  where  equally  dis- 
tant from  a  certain  point  within  it,  called  the 
Centre.  An  Arc  is  any  part  of  the  circum- 
ference. 

35.  The  Diameter  of  a  circle,  is  a  straight 
line  drawn  through  the  centre  and  termina- 

tifg  in   tho   ^-Vr'"— '''^'-"'>-   -''^    '•■•-'-    '"'^"         * - 

Q.  How  is  a  right  angle  formed  ?  4.  What  are  oblique  angles  ?  5.  ObtusF" 
6.  Acute?  6.  What  is  a  triangle  ?  7.  An  equilateral  triangle?  8.  An  Iso:^* 
celes?9.  Scalene?  10.  Right  angled  triangle  ?  11.  Obtuse  angled  triangk^^ 
12.  Acute  angled  triangle?  13.  What  are  the  names  of  the  sides  in  a  rifl;^ 
angled  triangle ?  14.  How  are  each  found?  16.  On  what  principle  is  ea 
operation  based  ?  15. 

*  Plane,  [L.  Planus.]  An  even  or  level  surface,  like  plain  in  common  langua/;e.    i*-J 
instrument  used  in  smoothing  boards. 


MENSURATION. 


267 


Chord  is  a  straight  line  shorter  than  the  diameter,  and  joins  the  ex- 
tremities of  an  arc.     The  arc  and  chord  together  form  a  Segment. 

36.  Every  chord  of  a  circle  divides  it  into  two  unequal  parts,  and 

every  diameter  into  two  equal  parts  called  Semicircles,  that  is,  half 

circles.    A  Radius  is  half  the  diameter,  or  a  right  line  drawn  from  the 

centre  to  the  circumference  ;  two  or  more  such  lines  are  called  Radii. 

Fig.  15.  Fig.  16.  Fig.  17. 


37.  A  Perpendicular  of  a  quadrilateral  or  triangle  is  a  straigh 
line  drawn  to  a  point  in  the  base  from  the  angle  opposite  to  that 
point,  as  the  dotted  lines  in  Fig.  15. 

38.  From  an  inspection  of  Fig.  15,  it  appears  that  the  area  or  sur- 
face of  a  Rhombus  or  Rhomboid,  is  equal  to  the  area  of  a  square 
or  parallelogram  of  the  same  length,  but  whose  breadth  is  its  perpen- 
dicular height. 

39.  A  Diagonal  is  a  line  that  passes  across  a  quadrilateral  from 
one  angle  to  its  opposite  one. 

40.  The  diagonal  of  every  parallelogram  divides  it  into  two  equal 
parts,  as  in  Fig.  16. 

41.  Hence  the  area  of  every  right-angled  triangle  is  just  half  as 
much  as  the  area  of  that  square  or  rectangle  whose  length  and  breadth 
are  equal  to  the  base  and  perpendicular  of  the  triangle. 

42.  Every  oblique-angled  triangle  may,  by  drawing  a  perpendicular 
to  its  base,  from  its  opposite  angle,  be  formed  into  two  right  angles. 

43.  Hence  the  area  of  every  obhque-angled  triangle  is  just  half  as 
much  as  the  area  of  that  square  or  rectangle,  whose  length  and 
breadth  are  equal  to  the  longest  side  and  perpendicular  of  the  triangle. 

RULES    FOR    FINDING    THE    AREAS    OF    SUPERFICES. 

44.  To  find  the  area  of  a  square  or  rectangle. — Multiply  the  length 
by  the  breadth. 

45.  To  find  the  area  of  a  rhomboid  or  rhombus. — Multiply  its 
length  by  its  perpendicular  height.  (See  38.) 

46.  To  find  the  area  of  a  right-angled  triangle. — Multiply  the  base 
by  half  the  perpendicular,  or  the  base  by  the  luhole  perpendicular,  and 
take  half  of  the  product.     (See  43.) 

47.  To  find  the  area  of  an  oblique-angled  triangle, — Multiply  the 
base  by  half  the  perpendicular,  draivn  from  the  opposite  angle.  (See 
41,  42.) 

48.  Or,  halve  the  sum  of  the  three  sides,  subtract  the  three  sides 

Q.  With  what  is  the  area  of  a  rhomboid  or  rhombus  compared  ?  38.  What  is 
a  diagonal?  39.  What  comparison  is  made  between  the  area  of  a  right-angled 
triangle  and  that  of  a  square?  41.  What  two  equal  divisions  may  be  made  of 
an  oblique-angled  triangle  ?  42.  What  is  the  inference  ?  43.  What  is  the  rule 
lor  finding  the  area  of  a  square  ?  44. — of  a  rhomboid  ?  45. — of  a  right-angled  tri- 
angle ?  46. — of  an  oblique-angled  triangle  ?  47,  48. 


268  ARITHMETIC. 

severally  from  this  half  sum,  multiply  the  four  results  together^  and 
find  the  square  root  of  the  product. 

COMMON    RULES    RESPECTING    CIRCLES. 

*  49.  The  diameter  is  to  the  circumference  nearly  as  7  :  22,  or 
more  accurately  as  113  :  355,  or  decimally,  as  1  :  3.14159  nearly; 
therefore, 

50.  To  find  the  circumference, — Either  multiply  the  diameter  hy 
22,  and  divide  by  7  ;  or  multiply  hy  355  and  divide  hy  113  ;  or  simply 
multiply  hy  3.14159. 

51.  To  find  the  diameter, — Reverse  the  foregoing  processes. 

52.  To  find  the  area  of  a  circle  : — Multiply  half  the  circumference 
by  half  the  diameter,  or  the  whole  circumference  by  half  the  radius  A 

53.  Suppose  one  field  is  60  rods  square,  and  another  contains  60 
square  rods  :  what  is  the  difference  in  their  areas  expressed  in  acres  1 

A.  22 A.  20rd. 

54.  If  a  site  for  a  house  is  in  the  form  of  a  square  with  150  feet 
front,  what  is  the  area  and  what  its  perimeter  ? 

A.  82y\«ysq.  rd. :  36T^rd.  round. 

55.  Suppose  you  contract  to  have  four  floors  made  at  S.75  per 
square  yard,  one  to  be  50  feet  square  and  the  other  three  each  20  feet 
square.  What  will  be  the  difference  between  the  cost  of  the  first  and 
that  of  the  others'?  A.  $108^ 

56.  What  will  be  the  length  of  each  side  of  a  square  formed  from 
an  area  of  10  acres?  A.  40  rods. 

57.  A  gentleman  has  two  valuable  building  lots,  one  containing  40 
square  rods,  and  the  other  00 ;  for  which  his  neighbor  offers  him  a 
square  field  containing  four  times  as  much  as  his  lots.  How  many 
rods  in  length  must  each  side  of  the  square  be  ?  .4.  20  rods. 

58.  What  are  the  contents  of  27  boards,  each  13  feet  long,  and  18 
inches  wide '?  A.  b2Q\. 

59.  Suppose  that  ten  boards  are  each  15  feet  long,  and  together 
contain  155  sq.  feet,  what  may  be  the  average  width  of  each  board  ? 

A.  \^\  ft. 

Q.  What  for  finding  the  circumference  of  a  circle  ?  50. — the  diameter  of  a 
circle  ?  51. — area  of  a  circle  ?  52. 

*  MORE   ACCURATE   ROLES. 

1.  To  find  the  circumference  of  a  circle  -.--Multiply  the  diameter  by  3.14159  ;  or  the  area 
by  12.56636217,  and  extract  the  square  root  of  the  product. 

2.  To  find  the  sideof  a  square  equal  to  a  given  circle -.—Multiply  the  diameter  by 
.886227,  or  the  circumference  by  .282094. 

3.  To  find  the  s^ide  of  an  equilateral  triangle  inscribed  in  a  circle  -.—Multiply  the  diam' 
eter  by  .866024,  or  the  circumference  6y.  2756616. 

4.  To  find  the  side  of  a  square  inscribed  in  a  circle :— Multiply  the  diameter  by  .707016, 
or  the  circumference  by  .225079. 

5.  To  find  the  area  of  a  circle  : — Multiply  the  square  of  the  diameter  by  .785398,  or  the 
square  of  the  circumference  by  .079577525. 

6.  To  find  the  diairieter  of  a  circle: — Multiply  the  circumference  by  .31831  ;  or  the  area 
by  1.273241,  and  extract  the  square  root  of  the  product. 

t  The  exact  ratio  of  the  diameter  to  the  circumference  of  a  circle  has  never  yet  been 
ascertained,  tbouijh  some  have  exhibited  an  approximation,  which  is  supposed  not  to 
vary  one  millionth  part  of  a  hair's  breadth,  in  the  sun's  distance  from  the  earth. 


MENSURATION. 


269 


60.  When  a  board  is  six  inches  wide,  how  long  must  it  be  to  con- 
tain 1  sq.  ft.]— 3  sq.  feet]— 7  sq.  ft.  1—12  sq.  ft.1     A.  Total  46ft. 

61.  If  a  road  150  miles  long  and  4  rods  wide,  would  cost,  when 
completed,  $2  per  square  rod,  what  would  the  land  cost  by  the  acre, 
allowing  the  cost  of  making  the  road  to  be  $2  per  rod,  (linear  meas- 
ure)? ^.  $240per  acre. 

62.  Suppose  a  square  has  an  area  of  7500  square  yards  ;  what  is 
the  breadth  of  a  walk  round  it  that  shall  take  up  just  two  thirds  of  the 
square]  A.  18.3013yd.  nearly. 

63.  If  a  rhomboid  is  50  feet  long  and  40  feet  wide,  what  is  its  area  ] 

A.  200ft. 

64.  If  one  rhombus  be  60  feet  long,  with  a  breadth  of  15  feet,  and 
another  45  feet  long,  with  a  breadth  of  20  feet,  what  is  the  difference 
in  their  areas  ]  A.  Nothing. 

65.  If  a  rhomboid  be  80  feet  long  and  60  feet  wide,  what  is  the  sum 
of  the  areas  of  the  two  ends  which  when  cut  off  will  leave  the  re- 
mainder in  the  shape  of  a  square  ]  A.  1,200  feet. 

66.  Herodotus  estimated  the  largest  and  most  remarkable  of  the 
Egyptian  pyramids  to  be  800  feet  square  at  its  base.  Now,  how  long 
a  road  4  rods  wide  would  occupy  as  much  land  as  the  base  of  the  pyra- 
mid] A.   Im.  6fur.  27yVTffrd. 

67.  Suppose  the  hypothenuse  and  perpendicular  of  a  right  angled 
triangle  be  50  and  30  feet,  what  is  the  area]  A.  600  feet. 

68.  If  the  sides  of  an  oblique  angled  triangle  be  40,  50,  and  80  feet, 
what  is  the  area]  A.  818 +  sq.  ft. 

69.  If  the  sides  of  a  triangle  be  16.6  ;  18.32,  and  28.6,  what  is  its 
area]  A.  143  nearly. 

70.  Suppose  a  field  has  one  right  angle,  and  its  hypothenuse  and 
base  are  100  and  80  rods ;  how  many  acres  does  it  contain  ]  A.  ISA. 

71.  Suppose  a  piece  of  land  in  the  form  of  a  right  angled  triangle, 
whose  angles  are  respectively  120  and  160  rods  :  what  is  the  area  ] 

A.  60 A. 

72.  What  is  the  circumference  of  a  circle  whose  diameter  is  15  ] 
(15x355^113.)  ^.47.12+. 

73.  What  is  the  diameter  of  a  circle  whose  circumference  is  350  ] 

A.  111.4  +. 

74.  What  is  the  area  of  a  circle  whose  diameter  is  24,  and  circum- 
ference 75  ]  ^.450. 

75.  What  is  the  area  of  a  circle  whose  diameter  is  24  ]  For  the 
method  of  solving  several  questions,  see  reference  from  49,  above. 

A.  452.3904+.* 

76.  What  is  the  area  of  a  circle  whose  circumference  is  75  ? 

A.  447.61875  +  . 

77.  If  the  diameter  of  a  circle  be  24,  what  is  the  length  of  one  side 
of  a  square  equal  to  the  circle  ?  ^.21 .269  + . 

78.  When  the  circumference  of  a  circle  is  75,  what  is  the  side  of  a 
square  equal  to  the  circle  ?  A.  2 1 .  157 + . 

23* 


270  ARITHMETIC. 

79.  What  is  the  diameter  of  a  circle  whose  areais  1151  A.  12.1 +  . 

80.  When  the  side  of  a  square  is  10.5  what  is  the  diameter  of  a 
circle  which  is  equal  to  the  square  1  A.  11.847  +  . 

81.  When  the  side  of  a  square  is  10.5  what  is  the  circumference  of 
a  circle  equal  to  the  square  ?  A.  37.224+. 

82.  When  the  diameter  of  a  circle  is  12,  what  is  the  area  of  a  semi- 
circle formed  from  that  circle  ?  A.  56.548  +. 

83.  Suppose  a  tract  of  land  is  5  miles  long  and  3  miles  wide,  what  is 
the  distance  round  a  square  of  an  equal  area?  A.  15m.  3fur.  37|rd. 

84.  If  a  field  be  48  rods  long  and  10  rods  wide,  what  will  be  the 
diameter  of  a  circle  of  equal  area  ]  Having  found  the  area  of  a  cir- 
cle, find  the  sum  of  the  areas  of  a  square  inscribed,  and  one  circum- 
scribed ;*  also  the  side  of  a  triangle  inscribed.  A.  Diameter,  24.721  + 
rods;  areas,  916.635    +  sq.  rd.;  side,  21.408+rd. 

85.  How  long  will  it  take  a  man,  going  at  the  rate  of  10  miles  in  2 
hours,  to  travel  round  an  area  of  256,000  acres,  laid  out  so  that  the 
circumference  shall  be  the  shortest  distance  possible  that  will  contain 
the  given  area?  ^.  14h.  lOfy^m. 

80.  A  circle  has  an  area  of  308  square  rods,  and  is  to  be  divided 
into  four  equal  concentric  ^  circles ;  what  will  be  the  width  of  each 
circular  part  ?  A.  9.9+rd.;  2.05  +  rd.;   1.57  +  rd.;  1.33+rd. 

87.  The  radii  of  two  concentric  circles  are  10  and  12  yards. 
What  is  the  area  included  between  them  ]         A.  138.22996 +yd. 

88.  There  is  a  meadow  of  10  acres  in  the  form  of  a  square,  and  a 
horse  tied  equidistant  from  each  angle  or  corner.  What  must  be  the 
length  of  the  rope  that  will  permit  the  horse  to  graze  over  every  part 
of  the  meadow?  ^.  28.284+rd. 

89.  In  the  midst  of  a  meadow  well  stored  with  grass, 
I've  taken  just  two  acres  to  tether  my  ass  ; 

Then  how  long  must  the  cord  be,  that,  feeding  all  round, 
He  mayn't  graze  less  or  more  than  two  acres  of  ground  \ 

A.  10.0925+rd. 


X 


90.  What  is  the  perimeter  of 
Fig.  18,  and  what  the  area  of  its  ^  /  \ 
lots,  a  and  6.?     A.  Perimeter,          ^^"^^  ^^-    '*' 
I01.4159rd.;  «==300  sq.  rd.;  h  = 
150  sq.  rd. 

91.  Since  an  acre  is  equal  to 
a  rectangle,  which  is  40  poles= 
10  chains  =  1,000  links  in  length, 
aftd  4  poles  =^1  chain  =  100  links 
in  breadth,  it  will  contain  1,000 
X  100  —  100,000  square  links, 
therefore — 

92.  If  the  linear  dimensions  be  expressed  in  links,  and  the  superji- 

*  The  diameter  of  the  circle  is  of  course  the  length  of  one  side  of  a  circumscribed 
eqnare.  ^    . 

1  CoNCBNTEic,  [L.  concentrico.}  Havmg  a  common  centre. 


/ 

Area. 

\ 

rd. 

10  rods. 

/    157 

0  7  9  5  sq. 

A 

a 

\ 

X 

B 

D 

\c 

MENSURATION. 


271 


cial  contents  he  found,  these  results,  when  divided  by  100,000,  or  luith 
five  figures  cut  off  towards  the  right,  will  give  the  number  of  acres  and 
parts  of  an  acre,  expressed  in  decimals. 

93.  The  length  of  a  rectangular  field  being  25  chains  8  links,  and 
its  breadth  14  chains  75  links,  what  number  of  acres  does  it  contain  1 
25  chains  8  links  =  2,508  links,  and  14  chains  75  links  =  l,475  links  j 
then  2,508  X  1,475=36. 99300  acres=36  acres  3  roods  38.88  poles. 

A.  36A.  3r.  38f|rd. 

94.  Find  the  area  of  a  square  field  whose  side  is  10^^  chains. 

A.  11  A.  4rd. 

95.  The  base  of  a  triangular  field  is  16  chains  3  poles,  and  its  per- 
pendicular 6  chains  2  poles  ;  what  number  of  acres  does  it  contain  ? 

A.  5A.  IR.  31rd. 

96.  What  is  the  length  of  the  side  of  a  square  field  comprising  2 
acres  and  4  poles'?  A.  4|-  chains. 

97.  Two  acres  of  land  are  to  be  cut  from  a  rectangular  field  whose 
breadth  is  2  chains  50  links,  by  a  line  parallel  with  either  end ;  what 
is  the  length  of  the  plot  1  A.  8  chains. 

Fig.  19. 


1     " 

\.              l 

/ 

c 
62c  f5l 

'      n 

h6e.  60l\/ 

CO 

i 
92e  SOl. 

' ■    ■                 1 

\l8cS0\J^         J 

iB, 

k 
U751 

98.  In  the  foregoing  plot,  the  figures  on  the  sides  of  each  lot  repre- 
sent so  many  chains  and  links.  The  road  that  extends  round  the 
whole  and  terminates  at  the  inner  circle  is  1  chain  in  width.  Re- 
quired with  these  data  the  number  of  acres  that  are  contained  in  the 
whole  figure  1 

Answers.— a  =  l\2'd.yi  be;  /!>=2,349.5c. ;  c=2,874  .4375c.;  d=^ 
2,085c.;  e=  1,077.625c.; /=- 1,271.875c.;  ^=2,945.59c.;  A=2,475.- 
9375c.;  i=3,214.375c.;  ;  =  829.576c.;  ^  =  5,690.3125c.;  square  with- 
out the  circles^  1,036.584c.  Total,  2,698A.  3sq.rd.;  without  the 
road  =18^  chains,  nearly. 

OFSOLIDS, 

CXII.  1.  A  Solid  is  any  thing  that  has  three  dimensions,  length, 
breadth,  and  thickness. 


272  ARITHMETIC. 

2.  A  Prism  is  a  solid  with  two  equal  and  parallel  bases  or  ends, 
and  sides  that  are  parallelograms.  The  sides  are  called  the  lateral 
surfaces. 

3.  Prisms  receive  particular  names,  according  to  the  figure  of  their 
bases,  as  triangular,  circular,  square,  pentagonal,  and  so  on. 

4.  A  Cube  is  a  solid  or  prism  bounded  by  six  equal  and  square 
eides. 

5.  A  Parallelopiped  is  a  prism  of  four  sides  and  two  ends,  whose 
length  is  more  than  its  breadth,  as  a  hewn  stick  of  timber. 

6.  A  Cylinder  is  a  round  prism,  whose  bases  or  ends  are  of  course 
circular,  like  a  round  column,  or  a  common  round  rule. 

7.  A  Pyramid  is  a  solid  whose  sides  taper  gradually  from  the  base 
to  one  common  point,  called  the  vertex  of  the  pyramid. 

8.  Pyramids  take  their  names  according  to  the  figure  of  their 
bases  or  ends,  as  circular,  triangular,  square,  and  so  on. 

9.  A  Cone  is  a  round  pyramid ;  of  course  its  base  is  circular,  as  a 
sugar  loaf,  if  it  comes  to  a  point  at  the  top. 

10.  A  Sphere  or  Globe  is  a  round,  solid  body,  that  has  a  centre 
equally  distant  from  every  part  of  the  surface,  as  an  orange. 

11.  The  Diameter  and  Periphery  of  a  sphere  are  the  same  as 
those  of  a  circle  of  equal  circumference.  A  Hemisphere  is  half  a 
globe. 

12.  A  Frustrum  or  trunk  of  a  pyramid  is  a  portion  of  the  solid 
next  the  base,  cut  oil'  so  that  its  base^  are  parallel.  The  other  part 
is  called  a  segment. 

13.  Thus  the  top  of  a  sugar  loaf  of  pyramidal  form,  cut  off  square, 
is  a  segment,  and  what  remains  is  a  frustrum. 

14.  The  Axis  of  a  solid  is  a  straight  line  passing  from  one  end  to 
the  other,  through  the  centre. 

15.  The  Axis  of  a  sphere  is  the  same  as  the  diameter  of  a  circle. 

16.  The  Altitude  or  Height  of  a  pyramid  is  the  perpendicular 
distance  from  the  apex  or  top  to  the  centre  of  the  base. 

17.  The  Slant  height  of  a  regular  pyramid  is  the  distance  from 
the  vrertex  to  the  middle  of  one  of  the  sides  of  the  base,  or,  if  it  be  a 
cone,  to  the  circumference  of  the  base. 

18.  A  Wedge  is  a  solid  that  has  a  rectangular  base,  two  triangular 
sides,  and  two  quadrilateral  sides  that  meet  in  an  edge,  as  the  wedge 
used  in  splitting  wood. 

19.  A  Prismoid  differs  from  a  prism  or  a  frustrum  of  a  pyramid 
only  in  having  its  ends  dissimilar. 

RULES    FOR    FINDING    THE    AREAS    AND    CONTENTS    OF    SOLIDS. 

20.  To  find  the  content  of  a  cube.— Ct/^e  either  side. ^ 

CXII  Q.  What  is  a  solid?  1.  Prism?  2.  Their  names?  3.  Cube?  4. 
Parallelopipgd?  5.  Cylinder?  6.  Py^^a^jid?  7.  Thei^r  names?  8  Cone?  8 
Sohere?  10  Diameter?  11.  Frustrum?  12.  Axis  of  a  solid ?  14.  Altitude 
of  a  solid  ?  16.  Wedge  ?  18.  Prismoid  ?  19.  What  is  the  rule  for  finding  the 
solidity  of  a  cube  ?  20.  ' 


MENSURATION.  273 

21.  To  find  the  solidity  of  a  prism,  or  cylinder,  as  of  round  timber. — 
Multiply  the  area  of  its  base  by  its  length* 

22.  To  find  the  area  of  a  prism  or  cylinder. — Add  together  the 
areas  of  the  different  sides  and  ends. 

23.  To  find  the  solidity  of  a  parallelepiped,  as  of  square  timber. — 
Multiply  the  length  by  the  breadth,  and  that  product  by  the  depth. 

24.  To  find  the  solidity  of  a  pyramid. — Multiply  the  area  of  the 
base  by  ^  of  its  height. 

25.  To  find  the  area  of  a  pyramid. — Multiply  half  the  slant  height 
by  the  perimeter  of  the  base  for  the  lateral  surface,  to  which  add  the 
area  of  the  base. 

26.  To  find  the  solidity-  of  a  sphere. — Multiply  the  cube  of  the  dia- 
meter by  .5236.  Or  multiply  the  square  of  the  diameter  by  \  of  the 
circumference.     Or  multiply  the  surface  by  -J  of  the  diameter. 

27.  To  find  the  area  or  surface  of  a  sphere. — Multiply  the  diameter 
by  the  circumference. 

28.  To  find  the  solidity  of  a  frustrum  of  a  pyramid. — Add  together 
the  areas  of  the  tivo  ends,  and  the  ?nean  proportional  between  these 
areas  ;  then  multiply  the  sum  by  ^  of  the  perpendicular  height. 

29.  To  find  the  area  of  a  frustrum  of  a  pyramid. — Add  together 
the  areas  of  the  sides  and  ends. 

30.  To  find  the  solidity  of  a  wedge. — Multiply  half  its  length  into 
the  length  and  breadth  of  its  base. 

31.  To  find  how  large  a  cube  may  be  inscribed  in  a  given  sphere, 
or  be  cut  from  it. — Divide  the  square  of  the  diameter  of  the  sphere  by 
3,  and  extract  the  square  root  of  the  qaotient. 

32.  The  side  of  a  cube  is  24  feet.     Required  its  content. 

A.  13,824  feet. 

33.  When  the  side  of  a  cube  is  25.5  inches,  what  is  the  solidity? 

A.  16,581.375  inches. 

34.  A  prism  is  20|  feet  high,  with  a  base  2\  feet  square.  Required 
its  content.  A.  104.025. 

35.  A  stick  of  timber  is  20  feet  long,  1  foot  8  inches  broad,  and  10 
inches  thick.     Required  its  solidity.  A.  ^7^-  feet. 

Q.  Of  a  cylinder?  21.  Of  square  timber?  23.  Of  a  pyramid?  2i.  Of  a 
sphere  ?  26.     Of  a  frustrum  of  a  pyramid  ?  28.     Of  a  wedge  ?  30. 

♦  The  dimensions  of  round  timberare  found  by  girting  the  tree  and  taking  J  of  the  girt 
for  the  side  of  a  square. 

When  the  tree  tapers  regularly^  the  girt  may  be  taken  at  the  middle,  or  at  both  ends, 
in  which  case,  half  their  sum  will  be  the  mean  girt.  When  the  timber  is  very  irregular, 
the  girt  may  be  taken  at  several  places,  equally  distant,  and  their  sum  divided  by  the 
number  of  girts ;  or  divide  the  tree  into  several  lengths,  according  to  its  irregularity,  find 
the  content  of  each  length  separately,  and  their  sum  will  be  the  whole  content  of  the  tree. 

In  measuring  oak  timber  with  the  bark  on,  a  deduction  of  777  or  j^  of  ^^e  circumfe- 
rence is  often  made  to  the  buyer ;  in  respect  to  elm,  beech,  ash,  &c.  the  deduction  is  less, 
because  the  bark  is  not  so  thick. 

General  Rule.— ilIu?<tpZy  the  square  of  the  quarter  girt,  or  the  square  of  \  of  the  mean 
circumference,  by  the  length  of  the  timber. 

This  method,  though  very  generally  used,  gives  the  content  about  i  less  than  that 
found  by  considering  the  tree  as  a  cylinder,  or  the  content  will  be  nearly  the  same  as  if 
the  tree  were  hewn  square. 


274  ARITHMETIC. 

36.  A  granite  column  is  50  feet  high,  and.  each  end  8  feet  in  cir- 
cumference.    Required  its  solidity.  A.  254.65  nearly. 

37.  A  stick  of  timber  is  19  feet  long,  with  trigonal  ends,  and  sides 
each  2  feet  wide.     Required  its  solidity.  A.  32.9sq.  ft.  nearly. 

38.  The  largest  Egyptian  pyramid  has,  according  to  Herodotus,  an 
altitude  of  800  feet,  and  a  square  base  whose  perimeter  is  3200  feet. 
Required  its  content.  A.  170,666,666|ft. 

39.  The  same  author  says  the  construction  of  the  above  pyramid 
occupied  100,000  men  nearly  20  years.  What  then  would  it  have 
cost  at  the  rate  of  $1|  per  day  for  each  man,  (he  boarding  himself,) 
allowing  26  working  days  to  the  month  ?  A.  $936,000,000. 

40.  What  is  the  solidity  of  a  globe  12in.  in  diameterl     A.  904,-'^  + 

41.  The  earth  is  about  25,000  miles  in  circumference.  Required 
its  solidity. 

A.   198,943,750sq.  m. :  263,857,570,390  cubic  miles,  nearly. 

42.  The  diameter  of  the  moon  is  about  2,180  miles.  What  is  ita 
solidity  ]  A.  5,424,017,475+  sq.  miles. 

43.  A  frustrum  of  a  pyramid  is  12  feet  high,  the  base  9  feet  square, 
and  the  other  end  6  feet  square.  Required  its  solidity.  (See  xcix. 
69.)  A.  684  s.  ft. 

44.  If  a  round  stick  of  timber  be  30  feet  long,  and  its  extreme 
peripheries  4ft.  6in.,  and  3ft,  9  in.,  what  is  its  solidity  \  The  ratio  of 
4ft.  6in.  to  3ft.  9in.  is  f,  or  which  is  more  convenient  \\.    (xcix.  76.) 

A.  40.0732-h  s.  ft. 

45.  One  end  of  a  rectangular  frustrum  is  60  feet  by  40,  the  other 
end  40  feet  by  30,  and  120  feet  in  length.     Required  its  solidity. 

A.  212,000  s.  ft.  nearly. 

46.  A  stick  of  hewn  timber  is  45  feet  long,  and  its  ends  are  24in. 
by  20,  and  30in.  by  24.  Required  its  solidity.    A.  186ys.  ft.  nearly. 

47.  If  the  base  of  a  wedge  be  30  by  8,  and  the  length  60,  what  is  its 
solidity]  ^.7,200. 

48.  Allowing  the  earth's  diameter  to  be  8,000  miles,  what  is  the 
side  of  the  largest  cube  that  can  be  inscribed  in  it  T 

A.  4,618.8  miles  nearly. 

49.  To  find  the  solidity  of  any  irregular  body  whose  dimensions 
cannot  be  ascertained. — Immerse  the  solid  in  a  regular  vessel  of 
water,  and  carefully  note  the  difference  betiveen  the  height  of  the 
water  before  the  immersion,  and  afterwards ;  for  the  requisite  dimen» 
sions,  with  which  proceed  according  to  previous  rules. 

50.  A  solid  immersed  in  a  vessel  18  inches  square,  raised  the 
water  9in.  Required  the  content  of  the  given  solid.  A.  1.6875  s.  ft. 

51.  A  boy  boasting  of  his  knowledge  of  arithmetic,  was  asked  by 
his  father  "  If  he  had  got  so  far  that  he  could  measure  a  brush  heap  1" 
Oh,  certainly,  says  he,  only  chop  it  up  fine  and  throw  it  into  the  cider 
vat,  and  it  is  done.  But,  rejoined  the  father,  suppose  the  vat  is  6ft. 
square,  and  the  cider  is  raised  by  the  brush  2  inches,  let  us  see,  after 
all,  if  you  can  calculate  the  contents  of  the  brush.  A.  6  s,  ft. 


MENSURATION. 


275 


SIMILAR      FIGURES. 

52.  When  two  figures  vary  in  size,  but  are  alike  in  shape  or  form ; 
they  are  called  Similar  Figures. 

53.  Similar  Figures  have  the  angles  of  the  one  equal  to  the  cor- 
responding angles  of  the  other,  each  to  each.  The  sides  opposite  to 
equal  angles  are  called  homologous  sides. 

RULES  OF  PROPORTION. 

54.  The  homologous  sides  of  all  similar  figures  are  proportional. 

55.  All  similar  figures,  whether  they  be  triangles,  quadrangles,  or 
polygons,  are  in  proportion  to  each  other  as  the  squares  of  their  ho- 
mologous sides. 

56.  The  circumferences  of  circles  are  in  proportion  to  each  other 
as  the  radii  or  diameters  of  the  circles.  The  same  is  true  of  the  arcs 
and  chords  of  similar  segments. 

57.  Circles,  or  their  areas  are  to  each  other  as  the  squares  of  their 
radii,  diameters  or  circumferences. 

58.  To  find  the  area  of  a  regular  polygon,  or  any  regular  figure  : 
Multiply  the  square  of  one  of  its  sides  by  the  area  of  a  similar  figure 
of  which  the  side  is  a  unit,  as  in  the  following : — 

TABLE  OF  REGULAR  POLYGONS. 


Names. 

Sides. 

rerpendiculars. 

Areas, 

Trigon, 

3 

0.2886752 

0.4330127 

Tetragon, 

4 

0.5000000 

1.0000000 

Pentagon, 

5 

0.6881910 

1.7204774 

Hexagon, 

6 

0.8660254 
1.0382601 

2.5980762 

Heptagon, 

7 

3.6339124 

Octagon, 

8 

1.2071069 

4.8284271 

Nonagon, 

9 

1.3737385 

6.1818242 

Decagon, 

10 

1.5388418 

7.6942088 

Undecagon, 

11 

1.7028439 

9.3656399 

Dodecagon, 

12 

1.8660252 

11.1961524 

Fig.  21. 


59,  Cubes,  globes,  and  all  similar  solids  are  to  each  other  as  the 
cubes  of  their  similar  dimensions.  -p^^  jo 

60.  In  figure  20  the  perpendicular  is  j?C 
40  rods,  and  the  base  30  rods  ;  what  is 
the  base  of  figure  21,  the  perpendicular 
being  28  rods ;  what  is  the  hypothenuse 
of  each  figure,  and  what  the  sum  of  the 
areas  of  both  1  [See  54,  55.]  A:BC 
=50rd.:ac=21rd.  :bc=25rd.  Areas 
5A.  2R.  14sq.  rd. 

Q.  What  are  similar  figures  ?  52.  Wherein  are  they  equal  ?  53.  Wliich  are 
the  homologous  sides  ?  53.  Which  sides  are  proportional  ?  54.  What  propor- 
tion have  similar  and  rectilineal  figures,  or  their  areas,  to  each  other  ?  55.  What 
proportion  have  the  circumference  of  different  circles  ?  56. 


276 


ARITHMETIC. 


61.  Wanting  to  know 
the  height  of  the  cathedral 
at  York,  I  measured  the 
length  of  its  shadow,  and 
found  it  to  be  200  feet. 
At  the  same  time  a  staff 
5  feet  long  cast  a  shadow 
of  4  feet :  required  the 
height  of  that  elegant  and 
magnificent  structure  : — 
[See  55.]     A.  250  feet. 


Fig.  22 


rrjMMi^  ' 


G2.  Being  desirous  of  finding  the  height  of  a  steeple,  I  placed  a 
looking  glass  at  the  distance  of  100  feet  from  its  base  on  the  horizon- 
tal plane,  and  walking  backwards  5  feet,  I  saw  the  top  of  the  steeple 
appear  in  the  centre  of  the  glass;  required  the  steeple's  height,  my 
eye  being  5  feet  6  inches  from  the  ground?     [See  54.]    A.  110ft. 


Fig.  23 


63.  When  the  sides  of  a  figure  are  each  25  rods,  what  would  be  its 
area  in  square  rods,  if  it  were — 


A  Pentagon  ? 
A  Hexagon  1 
A  Heptagon  1 
An  Octagon  1 


A.  1075.298  +  . 
A  1G23.797-1-. 
A.  2271. 195  +  . 
A.  3017.766  +  . 


A  Nonagon  1 
A  Decagon  ] 
An  Undecagon'? 
A  Dodecagon  1 


A.3863.640  +  . 
A.  4808.880  +  . 
yl.  5853.524+. 
^.6997.595+. 


64.  There  is  a  circle  whose  diameter  is  6  inches,  required  the  di- 
ameter of  one  two  times  as  large  1 — of  one  three  times  as  large  1 — of 
one  ten  times  as  large?  Ratios  2,  3,  and  10;  therefore,  6^x2=72; 
and-/72==8,485in.  the  diameter  of  one  two  times  as  large. 

A.  Total,  37.85in.+. 

65.  There  is  a  circle  with  a  diameter  of  12  inches, — what  is  the 
diameter  of  one  only  half  as  large  1 — I  as  large  1 — }  as  large  ? — I  as 
large  ?— |  as  large  ?— f  as  large  ?  A.  Total,  44.938in.  + 

66.  If  113.097  be  the  area  of  a  given  circle,  what  will  be  the 
area  of  one  4  times  as  large,  and  the  area  of  one  whose  diameter  is  4 
times  as  large  1  (Retain  3  decimal  figures.)     A.  452.388 :  1809.652. 

67.  If  a  ball  3  inches  in  diameter  weigh  41b.,  what  will  a  ball  of 
the  same  metal  weigh,  whose  ctlameter  is  6in  ?  3^ :  6^ :  41b.    A.  321b. 

Q.  What,  the  area  of  circles  ?  57.  What,  cubes  and  all  similar  figures  ?  59, 
How  is  the  area  of  a  regular  polygon  found  ?  58. 


GAUGING 


277 


68.  There  are  two  little  globes,  one  of  them  is  1  inch  in  diameter 
and  the  other  two  inches ;  how  many  of  the  smaller  globes  will  make 
one  of  the  larger?  A.  8  gbbes. 

69.  If  the  diameter  of  the  planet  Jupiter  is  12  times  as  great  as 
the  diameter  of  the  earth,  how  many  globes  of  the  earth  would  it  take 
to  make  one  as  large  as  Jupiter]  A.  1728  globes. 

70.  If  the  sun  be  1,000,000  times  as  large  as  the  earth,  and  the 
earth  8,000  miles  in  diameter,  what  is  the  diameter  of  the  sun  1 

A.  800,000  miles. 


GAUGING. 


CXIII.  1.  Gauging  is  the  process  of  ascertaining  the  capacity  of 
any  regular  vessel,  in  bushels,  gallons,  &c. 

2.  The  ale  gallon  contains  282  cubic  inches. 

3.  The  wine  gallon  contains  231  cubic  inches. 

4.  The  bushel  contains  2,150.4  cubic  inches. 

5.  A  cubic  foot  of  pure  water  weighs  1,000  ounces =62|  pounds 
avoirdupois. 

6.  To  find  what  weight  of  water  may  be  put  into  a  given  vessel. — 
Multiply  the  cubic  feet  by  1000 /or  the  ounces,  or  by   G2-|-  for  the 

vounds,  avoirdupois. 

7.  What  weight  of  water  can  be  put  into  a  cistern  7^  feet  square  ? 

A.  26,3671b.  3oz. 

8.  What  weight  of  water  will  fill  a  circular  fish  pond  that  is  15 
rods  in  circumference,  and  has  an  uniform  depth  of  4  feet  \ 

A.  G09T.  6cwt.  2qr.  41b.  lOoz.  12dr. 

9.  To  find  the  number  of  gallons  or  bushels  that  a  given  vessel 
may  contain. — Calculate  the  content  in  inches,  which  divide  by  282 
for  the  ale  gallons  ;  by  231  for  the  wine  gallons,  and  by  2,150.4  for 
the  bushels. 

10.  How  many  barrels  of  ale  will  a  vat  8  feet  square  hold  ? 

A.  87bl.  SyVrgal. 

11.  What  will  the  oats  cost  at  62^  cents  a  bushel,  that  will  fill  25 
bins,  12  of  which  are  cylindrical,  being  18  feet  in  circumference,  and 
7  feet  deep ;  and  the  rest  7  feet  square  1  A.  $3327. 18  + 

12.  A  cellar  20  feet  long,  15  feet  wide,  and  8  feet  deep,  became, 
during  a  heavy  rain,  filled  with  water.  What  would  be  the  expense, 
when  labor  is  Slj  per  day  of  8  hours,  of  removing  the  water,  allowing 
that  one  man  can  empty  three  buckets,  in  2  minutes,  each  bucket  to 
hold  2 1  gallons,  (wine  measure)?  A.  $11,688+. 

13.  To  find  the  number  of  gallons  in  a  cask,  or  to  gauge  it : — 

CXIII.  Q.  What  is  Gauging  ?  1.  What  is  the  number  of  cubic  inches  in 
an  ale  or  wine  gallor  T  2,  3.  In  a  bushel  ?  4.  What  is  the  rule  for  finding  what 
weight  of  water  may  be  put  into  a  given  vessel  ?  6.  What  weight  of  water 
may  be  put  into  a  vessel  1  foot  square  ?  Into  one  2  feet  square  ?  Into  one  3 
feet  deep  and  2  feet  square  at  each  end  ?  What  is  the  rule  for  ascertaining  the 
capacity  of  vessels?  9.  ^j^ 


278  ARITHMETIC. 

RULE. 

14.  Take  the  dimensions  in  inches,  viz.,  the  diameter  of  the  hung 
and  head,  and  i)ie  length  of  the  cask,  and  find  the  difference  betiveen 
the  bung  and  the  head  diameter. 

15.  If  the  staves  of  the  cask  be  much  curved  betiveen  the  bung  and 
the  head,  multiply  the  difference  {found  above)  by  .7 ;  if  not  quite  so 
much  curved,  by  .65 ;  if  they  bulge  yet  less,  by  .6 ;  a7id  if  they  are  al- 
most straight,  by  .55;  add  the  product  to  the  head  diameter;  the  sum 
ivill  be  the  mean  diameter,  by  which  the  cask  is  reduced  to  a  cylinder. 

10.  Square  the  mean  diameter  thus  found,  then  multiply  it  by  the 
length ;  divide  the  product  by  359  for  ale  or  beer  gallons,  and  by  294 
for  wine. 

17.  There  is  a  certain  cask,  whose  bung  diameter  is  35  inches, 
head  diameter  27  inches ;  and  length  45  inches.  Required  its  capa- 
city in  ale  and  wine  gallons. 

18.  Thus,  35-27=  8 X. 7=  5.6+27=  32.6x32. 6=  1062.  76X45 
=  47824.20-^359  and  294=^.  133.21gal.  ale,  and  162.66gal.  wine. 

19.  What  is  the  content  of  a  cask  in  wine  and  ale  gallons,  whose 
bung  diameter  is  36  inches,  head  diameter  30  inches,  and  length  48 
inches?  A.  153.65  + ale  gal. ;  187.62+ wine  gal. 

20.  To  find  the  capacity  of  a  vessel,  which  is  in  the  form  of  a  lower 
frustrum  of  a  cone,  that  is  round,  and  larger  at  one  end  than  at  the 
other. 

RULE. 

21.  To  the  product  of  the  diameters  add  ^  of  the  square  of  their 
difference ;  which  result  multiply  by  the  height,  and  divide  as  above 
directed. 

22.  What  is  the  capacity  both  in  wine  and  ale  gallons,  of  a  tub  40 
inches  in  diameter  at  the  top,  32  inches  at  the  bottom,  and  its  per- 
pendicular height  48  inches'? 

A.  174  ale  gal.  nearly+  ;  212.46+  wine  gal. 


TONNAGE    OF    VESSELS. 

CXIV.  1.  Ship  Carpenters' Rule.  For  single-decked  vessels, 
multiply  the  breadth  of  the  main  beam,  the  depth  of  the  hold,  and  the 
length  together,  and  divide  the  product  by  95 ;  the  quotient  tvill  he 
tons.  For  double-decked  vessels,  take  one  half  of  the  breadth  of  the 
main  beam  for  the  depth  of  the  hold,  with  which  proceed  as  before. 

2.  A  single-decked  vessel  is  80  feet  long,  25  feet  broad,  and  12 
feet  deep.     Required  its  tonnage.  A.  252y|  tons. 

3.  Required  the  tonnage  of  a  double-decked  vessel,  whose  length 
is  80  feet,  and  breadth  26  feet.  A.  284|f. 

4.  Government  Rule.  "If  the  vessel  be  double-decked,  take  the 
length  thereof  from  the  fore  part  of  the  main  stern,  to  the  after  part 

Q.  What  is  the  rule  for  ascertaining  the  capacity  of  casks  ?  14,  15,  IG. 
CXIV.    Q.  What  is  the  rule  employed  by  ship  carpenters  in  estimating  the 
tonnage  of  vessels  ?    What  is  the  government  rule  ?  4. 


EXCHANGE.  279 

of  the  stern  post,  above  the  upper  deck  ;  the  breadth  thereof  at  the 
lowest  part  above  the  main  wales,  half  of  which  breadth  shall  be  ac- 
counted the  depth  of  such  vessel,  and  then  deduct  from  the  length  ^ 
of  the  breadth  ;  multiply  the  remainder  by  the  breadth,  and  the  pro- 
duct by  the  depth,  and  divide  this  last  product  by  95,  the  quotient 
whfereof  shall  be  deemed  the  true  contents  or  tonnage  of  such  ship  or 
vessel ;  and  if  such  ship  or  vessel  be  single-decked,  take  the  length 
and  breadth  as  above  directed,  deduct,  from  the  said  length,  |  of  the 
breadth,  and  take  the  depth  from  the  under  side  of  the  deck  plank  to 
the  ceiling  in  the  hold,  and  then  multiply  and  divide  as  aforesaid,  and 
the  quotient  shall  be  deemed  the  tonnage." 

5.  A  single-decked  vessel  is  75  feet  long,  breadth  25  feet,  and 
depth  12  feet.     Required  the  government  tonnage  of  it. 

A.    ISOf'^  tons. 

6.  A  double-decked  vessel  is  97  feet  in  length,  and  breadth  30  feet. 
Required  the  government  tonnage.  A.  374y\. 

7.  What  is  the  government  tonnage  of  a  double-decked  vessel, 
with  a  keel  of  115  feet  and  32.6  breadth  \  Jl.  533.841  +  tons. 

8.  What  is  the  government  tonnage  of  a  single-decked  vessel  hav 
ing  80  feet  keel  and  35  feet  breadth  at  the  beam,  and  14  feet  deep  in 
the  hold!  ^.  304yV 

9.  What  must  have  been  the  government  tonnage  of  Noah's  ark, 
the  length  of  which  was  300  cubits,  the  breadth  by  the  midship  beam 
50  cubits,  and  the  depth  in  the  hold  30  cubits,  allowing  the  cubit  to  be 
22  inches'! 

As  the  ark  was  differently  constructed  from  modern  vessels, 
we  must,  although  it  had  more  than  one  deck  ["  with  lower,  second 
and  third  stories  shalt  thou  make  it"]  calculate  its  tonnage  by  the 
rule  for  single  decked  vessels.  A.  26,269-1-^. 


EXCHANGE.* 

CXV.  1.  Exchange  is  the  method  by  v,'hich  v*'e  find  what  sum 
of  money  of  one  country,  is  equivalent  to  a  given  sum  of  another,  ac- 
cording to  some  settled  rate  of  commutation. 

2.  The  Course  of  Exchange  is  the  quantity  of  money  of  one 
country,  wh^ch  is  given  for  a  fixed  sum  of  another ;  the  former  is 
called  the  uncertain  price,  the  latter  the  certain  price. 

3.  The  Par  of  Exchange  may  be  considered  either  as  intrinsic  or 
commercial. 

4.  The  Intrinsic  Par  is  the  value  of  the  money  of  one  country 
compared  with  that  of  another,  both  with  respect  to  its  weight  and 
fineness. 

CXV.  Q.  What  is  Exchange?  1.  The  Course  of  Exchange?  2.  The  Par 
of  Exchange?  3.     The  Intrinsic  Par?  4. 

*  Most  of '.vhal  was  said  on  Notes  [Lxxxi.]  and  Banking  [lxxxv.]  in  respect  to 
Ihc  obligatiuns  of  the  parties,  is  applicithlc  to  Bills  of  Exciiauge.  The  subject,  how- 
evar,  ha.j  so.iic  pocu.iarais-^  \vhu;ii  it  iiny  bei<roper  to  notice. 


280  ARITHMETIC. 

5.  The  Commercial  Par  is  when  a  comparison  is  made  in  respect 
to  the  market  prices  of  the  metals. 

7.  Agio  is  the  difference  between  bank  notes  and  current  coins ;  it 
also  means,  in  places  where  foreign  coins  are  current,  the  difference 
between  the  actual  value  of  such  coins,  and  their  current  value  as 
fixed  by  government.  • 

8.  Usance  is  the  usual  time  at  which  bills  are  drawn  between  cer- 
tain places,  as  one,  two,  or  three  months  after  date,  or  sight. 

9.  When  the  course  of  exchange  in  any  place  runs  low,  it  is  favor 
able  to  that  place,  that  is,  to  its  buyers  or  remitters,  but  unfavorable 
to  drawers  and  sellers. 

10.  For  when  the  exchange  is  against  a  place,  it  is  an  object  with 
remitters  to  pay  their  foreign  debts  in  specie ;  the  exportation  of  which 
is  considered  a  national  disadvantage. 

11.  A  Bill  of  Exchange,  or  Draft,  is  a  written  obligation  to  pay 
a  certain  sum  of  money  at  a  specified  time,  and  is  either  Foreign  or 
Inland. 

12.  An  Inland  Bill  or  Draft  is  one  payable  in  the  same  country 
where  it  is  drawn. 

13.    FORM    OF   A   DRAFT. 

$2,000.  Hartford,  Jan.  15,  1840. 

Two  months  after  date,  pay  to  the  order  of  Messrs.  Spalding 
and  Storrs,  two  thousand  dollars,  value  received,  with  or  without  fur- 
ther advice  from  me.  D.  Burgess. 
Messrs.  J.  (Sf  J.  Harper,    > 
Booksellers,  New  York.  5 
14.  A  Foreign  Bill  is  an  order  from  a  person  of  one  country  to  a 
person  of  another  country,  requesting,  him  to  pay  to  a  third  person,  or 
to  his  order,  either  on  demand  or  at  a  specified  time. 

15.  form  of  a  foreign  bill. 
jC500  sterling.     -  Boston,  June  18,  1840. 

At  ninety  days  sight,  pay  to  Rufus  Smallet,  or  order, 
five  hundred  pounds  sterling,  value  received,  and  charge  the  same  to 
the  account  of  Norman  Wells. 


3^0  Messrs.  Stimpson  <Sf  Co.  ) 


Merchants,  London. 

IG.  It  seems  that  Bills  of  Exchange' are  the  same  thing  in  reality 
as  a  common  order,  the  only  distinctions  being  in  respect  to  the  places 
of  residence  of  the  parties,  and  the  ceremony  of  collection. 

17.  In  a  Bill  of  Exchange  there  are  usually  concerned  four  per- 
sons, viz. 

Q.  Tho  Commercial  Par?  5.  What  is  Agio?  7.  What  is  Usance?  8. 
When  is  the  course  of  exchange  favorable  to  any  place  ?  9.  What  is  the  ex- 
planatian?  10.  What  is  a  Bill  or  Draft?  11.  An  Inland  Bill ?  12.  What  is 
the  for.n  of  a  Draft?  13.  What  is  a  Foreign  Bill?  14.  Its  form?  15.  W^hat 
similarity  has  a  bill  to  a  draft  ?  16.  How  many  persons  are  generally  concerned 
in  a  Bill  of  Exchange  ?  17. 


EXCHANGE].  281 

18.  The  Drawer,  to  whom  the  bill  is  made  payable,  and  who  is  also 
called  the  Maker  and  Seller  of  the  bill. 

19.  The  Drawee,  on  whom  the  bill  is  drawn,  and  who  is  also  called 
the  Acceptor,  if  he  accepts  it,  which  is  done  by  writing  his  name  either 
at  the  bottom  or  across  the  back,  with  the  word  acccjded  over  it,  by 
which  act  he  becomes  responsible  for  its  payment. 

20.  The  Buyer,  who  purchases  it,  and  who  is  also  called  the  Taker 
and  Remitter. 

21.  The  Payee,  to  whom  it  is  ordered  to  be  paid  by  indorsement, 
and  who  may  pass  it  to  any  other  person  in  the  same  manner. 

22.  An  indorsement  may  be  either  blank  or  special. 

23.  A  blank  indorsement  is  merely  the  name  of  the  person  written 
on  the  back  of  the  bill,  which  then  becomes  transferable,  like  any  ar- 
ticle of  merchandize. 

24.  A  special  indorsement  has  with  the  name  a  specification  direct- 
ing to  whom  the  bill  shall  be  paid  ;  corisequently,  the  holder  must  put 
his  own  name  on  it  before  he  can  negotiate  the  bill. 

25.  Special  indorsements  are  preferable  when  bills  are  to  be  trans- 
mitted a  great  distance ;  for,  should  they  fall  into  improper  hands, 
the  indorser's  name  must  be  forged  before  they  can  be  negotiated, 
and  consequently  fraud  or  imposition  is  prevented  as  much  as  possible. 

26.  All  the  indorseis  are  severally  responsible  to  the  holder  of  the  bill, 
and  must  pay  it  in  case  the  acceptor  fails  to  do  so  at  the  proper  time. 

27.  A  Set  of  Exchange  implies  that  there  are  two  or  more  bills 
drawn  at  the  same  time,  and  of  the  same  tenor  and  date. 

28.  These  bills  are  drawn  for  the  purpose  of  being  transmitted  by 
different  ships,  or  posts,  as  a  security  against  accidents  and  delays, 
and  when  one  of  them  is  accepted  and  paid,  the  others  are  of  no  fur- 
ther use. 

29.    FORM    OF    A    BILL   IN    A   SET    OF   EXCHANGE. 

$4,500rV5^.  London,  June  5,  1840. 

Sixty  days  after  sight  of  this,  my  first  of  Exchange, 
(second  and  third  unpaid,)  pay  to  the  order  of  James  Cornwall,  Esq. 
four  thousand  five  hundred  dollars  and  seventy-five  cents,  value  re- 
ceived. John  Smith. 
To  Messrs.  Williams  <Sf  Co.} 
Merchants,  New  York.      S 

30.  Quotations  are  the  lists  of  the  courses  of  exchange  which  are 
transmitted  from  one  country  or  place  to  another,  for  the  brokers  and 
others,  and  are  made  in  the  following  manner. 

31.  Thus,  "London  on  the  United  States  2  percent,  advance" 

Q.  What  is  the  Drawer  ?  18.  The  Drawee  ?  19.  The  Buyer  ?  20.  The 
Payee?  21.  What  is  an  Indorsement?  22.  A  Blank  Indorsement?  23.  A 
Special  Indorsement  ?  24.  Why  is  the  latter  preferable  in  any  case  ?  25. 
What  liabilities  do  indorsers  incur  ?  26.  What  is  a  Set  of  Exchange  ?  27. 
What  is  their  use  ?  28.  Give  an  example.  29.  What  are  Quotations  ?  30. 
What  is  meant  by  "  London  on  the  United  States  at  2  per  cent,  advance  ?"  31. 
Repeat  the  table  of  sterling  money. 
24* 


2S2  ARITHMETIC 

ftieans,  that  a  bill  drawn  by  a  merchant  of  London  on  his  broker  in 
the  United  States  is  worth  2  per  cent,  more  than  the  par  value  in  the 
latter  place.  The  par  value,  we  have  seen,  is  4s.  6'd.  sterling  foT 
every  dollar  in  federal  money. 

TABLES  OF  FOREIGN  CURRENCIES. 

32.  These  tables  show  the  principal  denominations  of  the  curren- 
cies of  those  nations  with  which  the  United  States  has  the  most  in- 
tercourse, together  with  the  nominal  Or  fat  valuei  of  each,  expressedt 
in  federal  money,  (i.xxvi.  3,  4,  5.) 

33.    GREAT    BRITAIN    AND    IRELAND.* 

4farthings  sterling  ==1  penny  X  12=^1  shillingx20=jei==$4.444  +  . 
4s.  6d.  sterlings $l=£:/o.     21  shillings =1  guinea       =$4.666f. 

34.    BRITISH   AMERICA. 

5  shillings =$ I,  or  j£:|.     20  shillings ^j^l    -    -    -    -     =$4.00* 
In  Jamaica  6s.  8d.=$l  or  £i  and  £1 =$3.00. 

35.    SPAIN    AND    HER    DEPENDENCIES SPANISH   AMERICA. 

10  reals  vellon=l  real  plate  x  10  =  1  dollar  plate    -    -     =$1.00. 

36.    HOLLAND   AND    BELGIUM. 

100  Flemish  cents  =  1  florin  or  guilder     -    -    -    .^    *     =$.40. 
2|  florins=$l.     20s.  riemish=jei  Flemish    -    -    -     =$2.40 

37.    PORTUGAL. 

1,000  rees=l  milree =$1.24. 

38.    FRANCE. 

10  centimes =1  decimex  10=1  franc  -    -----     =$.18|. 

39.    RUSSIA. 

10  copecks=l  grievenerx  10  =  1  rouble  -    -    -    -    -     =$.75. 

40.    PRUSSIA. 

12  pfennings=l  good  groschenx24=  1  rix  dollar  -    -     =$.681. 

41.    SWEDEN    AND    NORWAY. 

12  runstycken=l  skillingx  48=1  rix  dollar      -    -    -     =$1.05. 

42.    DENMARK. 

16  skillings=:l  mark X  0=1  rigsbank  dollar  -    -    -    -     =$.53. 

43.    GENOA   AND    LEGHORN. 

12  denari  de  pezza=  1  soldo  de  pezzax20=  1  pezza  -     =$.90. 
12  denari  de  lira=  1  soldo  de  lira  x  20=  1  lira    -    -    -     =$.15f .    , 

44.    ROME. 

10  bajocchi=  1  paolox  10=  1  scudo  or  Roman  crown      =$1.00* 

45.    lifAPLES. 

10  grani=l  carlinoxlO=l  ducato =$.80. 

Q,  How  many  dollars  and  cents  in  £\  sterling?  33,  In  1  guinea?  33.  Irt 
4s.6d.  ?  33.  What  is  the  ratio  of  the  dollar  to  the  pound  sterling  ?  33.  How 
many  dollars  make  £\  in  British  America?  34.  What  sum  in  federal  money  is 
equal  to  1  dollar  plate  ?  35 — to  1  florin  or  guilder  ?  36 — to  1  milree  ?  37 — to  1 
franc  ?  38— to  1  centime  ?  38— to  1  rouble  ?  39— to  1  rix  dollar  ?  40— to  1  Roman 
crown?  44— to  1  rupee?  50— to  1  tale?  51— to  1  mace?  51— to  1  guilder?  52« 

*  Formerly  the  Irish  pound  was  about  $4.10^. 


COINS.  283 

46.  MALTA. 

20  grani=  1  taro  x  12=  1  scudo=$.40.    30  tari=  1  pezza=$1.00. 

47.  SICILY. 

20  grani=  1  taro  X  12=  1  scudo  or  Sicilian  crown  -    -    =  $.95. 
30  tari=  1  oncia      - =$2.40 

48.  VENICE. 

10  millesimi=  1  centesimo  x  10=  1  lira  de  Austria  -    -    =  $.  16^. 

49.    AUSTRIA. 

4  pfennings=  1  crewtzer  x  60=  1  florin  or  guilder  -    -    =  $.48 
1^  florin  or  guilder  =1  rix  dollar  of  account  -    -    -    -    =$.72 

50.    EAST    INDIES. 

8  pices=l  annaXlO=l  rupee   -- =$.55| 

61.    CANTON,    (china.) 

10  cash=  1  candarine  x  10=  1  mace  x  10=  1  tale   -    -    =  $1.48. 

52.    BATAVIA,    (JAVA.) 

3  dubbles=  1  skilling  x  4=  1  florin  or  guilder      -    -    -    =  $.40. 

53.    SURINAM,    BERBICE,    DEMARARA,    AND   ESSEQUlBO. 

8  doits=l  Stiver  X  20=1  guilder =$.33^. 

54.    ST.    DOMINGO    AND    HAYTI. 

The  currency  is  federal  money,  as  in  the  United  States. 


COINS. 

CXVI.  1.  When  the  American  mint  was  established,  in  1770, 
pure  gold  was  worth  a  trifle  more  than  15  times  as  much  as  an  equal 
quantity  of  pure  silver. 

2.  When  gold  had  become  more  valuable,  being  worth  about  16 
times  as  much  as  the  same  quantity  of  silver,  congress  passed  the 
act  of  1834,  by  which  it  was  ordered  that  our  gold  coins  should  con- 
tain 15^  grains  less  of  pure  gold  than  formerly,  in  order  that  the  eagle 
should  not  be  worth  any  more  than  10  dollars,  its  original  value. 

3.  Hence  we  see  why  old  eagles  are  worth  more  than  new  ones ; 
also,  why  foreign  coins  pass  for  more  than  formerly. 

4.  The  eagle  of  the  old  coinage  was  to  be  of  the  value  of  10  dollars, 
and  to  contain  270  grains  of  standard  gold,  or  247|  grains  of  pure 
gold,  and  22?  grains  of  alloy.  The  standard  was  22  carats  or  j^  fine 
gold,  and  j-^  alloy. 

5.  By  the  act  of  1834,  the  eagle  is  to  contain  258  grains  of  stand- 
ard gold,  or  232  grains  of  pure  gold  and  26  grains  of  alloy,  and  to  be 
of  the  value  of  10  dollars.     Old  eagles  are  worth  about  $10,665. 

6.  Th»  dollar  is  to  possess  the  same  value  as  the  Spanish  milled 

CXVI.  Q.  What  was  the  comparative  value  of  gold  and  silver  in  1790?  1. 
What  has  occasioned  a  difference  in  the  value  of  the  eagle  1  2,  3.  What  were 
the  component  parts  of  the  old  eagle  ?  4.  What  are  the  component  parts  of  the 
new?  5.    What  of  the  dollar?  6. 


2d4 


ARITHMETIC. 


dollar,  and  to  contain  31 1}  grains  of  pure  silver  and  44f  grains  of 
pure  copper,  forming  a  standard  of  416  grains. 

7.  The  price  established  at  the  mint  for  pure  silver  is  $0.20936 
per  grain. 

8.  The  cent  must  be  of  the  value  of  the  one  hundredth  part  of  a 
dollar,  and  contain  208  grains  of  copper. 

9.    TABLE. 
Value  of  the  principal  Coins   according  to  the  Act  of  Congress,  1834. 
UNITED    STATES. 

Eagle,  p.^  -  -  SIO.OOO 
Eagle,  (old)  -        -       10.665 

DoUar,  s.^  p.  -        -        -     1.000 

GREAT    BRITAIN. 

Guinea,  (21s.)  p.  c.  -      $5,075 

7  shilling  piece,  c.  -  -  1.698 
Sovereign,  (20s.)  p.  c.  -        4.846 

Crown,  (5s.)  s.  p.     -  -     1.100 

Half  Crown,  s.    -  -           .550 

Shilling,  s.      -         -  -       .220 

Sixpence,             -  -           .110 

fRANCE. 

Double  Louis,  p.  c.  -  ^9.697 

Louis='  p.  c.          -  -         4.846 

Double  Louis,' c.     -  -    9.153 

Louis,'  c.  -  -  -  4.576 
Napoleon,  (20  francs)  c.  -    3.851 

Double  Napoleon,  c.  -        7.702 

Guinea,  p.  c.  -         -  -    4.655 

5  Franc  piece,  s.  c.  -          .930 

2  Franc  piece,  S.     -  -       .372 

Franc,  s.      -        -  -           .186 

60  centimes,  s.        -  -       .093 

25  centimes,  s.     -  -        .04^i| 

SPANISH  POSSESSIONSj  MEXICO, 
COLOMBIA. 

Doubloons,  p.  c.  .  -  $16,028 
Patriot  Doubloons,  c  -  15.535 
Pistole,  c.  -        -  3.884 

Coronilla  (dollar)  c  -  .983 
Dollar,  s.  p.  c.  -        -  1.000 


PORTUGAL  AND  BRAZIL. 

Dobraon,  p,  c.     -        -    $32,706 


Dobra,  p.  c.      - 

17.301 

Johannes,  c. 

17.064 

Moidore,  c.       -        - 

6.557 

Piece  of  16  Testoons,^  c. 

82.121 

Old  Crusado,  (400  rees)  c 

.      .588 

New  Crusado,  (480  rees) 

c.    .635 

Milree,(1775)c.    - 

.730 

HOLLAND    AND    BELGIUM. 

Gold  Lion,  or  14  florin 

piece. 

$5,046 

Ten  florin  piece,^    - 

4.020 

Florin,  s.     -         -        - 

.400 

RUSSIA. 

Ducat,  (1796)     - 

$2,297 

Ducat,  (1763) 

2.667 

Gold  Ruble,  (1756)     - 

.967 

Gold  Ruble,  (1799) 

.737 

Gold  Polten,  1777)      - 

.355 

Imperial,  (1801)  p. 

7.829 

Half  Imperial,  (1801) 

3.918 

Half  Imperial,  (1818) 

3.933 

Ruble,  s.     - 

.750 

PRUSSIA. 

Ducat,  (1748)    - 

$2,270 

Ducat,  (1787) 

2.267 

Frederick,  double,  (1769) 

7.975 

Frederick,  double,  (1800) 

7.950 

Rix  Dollar,    - 

.690 

SWEDEN    AND    NORWAY. 

Ducat,       -        -         - 
Rix  dollar,  s. 

HAMBURG. 

Ducat, 

Crown  dollar,  s. 


$2,235 
1.070 

$2,800 
1.090 


Q.  What  are  the  component  parts  of  the  cent  ?  8.  What  does  pure  silver 
bring  at  the  mint?  7.  What  is  the  difference  in  value  between  the  eagles 
coined  before  1834  and  those  coined  since  ?  9.  What  is  the  value  of  the 
guinea,  sovereign,  and  crown  of  England  ? — of  the  double  Louis,  Louis,  Napo- 
leon, and  franc  of  France  ? — of  Spanish  doubloons  imd  pistoles  ?       • 

1.  The  letter  s.  stands  for  silver  coins,  p.  for  shares  in  proportion,  c.  for  those  ren- 
dered current  by  act  of  Congress,  1834.  Coins  whose  metal  is  not  indicated  by  any 
letter  are  gold  coins. 

2.  Coined  since  1786.  3.  Coined  since  178G.  4.  Coined  since  1772.  5.  Or  1600  rees. 
6.  Coined  since  1820. 


REDUCTION    05*    FOREIGN    CURRENCIES. 


285 


DENMARK. 

Ducat,  current,    - 
Ducat,  specie, 
Christian  d'or,    - 
Rigsbank  dollar,  s. 

GENOA    AND 

Sequin, 
Francescone,  s. 

NAPLES. 

0  Ducat  piece  (1783)    - 

2  Ducat  piece,  (Sequin) 

3  do.  or  Oncetta,  s. 
Ducat,  s.        -         - 

SICILY. 

Ounce,  (1751)      - 
Double  ounce,  (1758) 
Ducat, 

ROME. 

Sequin,  (1760)    - 
Scudo  of  Republic, 
Scudo,  or  Roman  crown,  s 

VENICE. 

Sequin,     -        -        - 
Ducat,  effective,    - 

TURKEY. 

Sequin,  (1818)    - 
Spanish  dollar, 

MALTA. 

Double  Louis,    - 


-      $1,812 

-  2.670 
4.021 

-  .530 

LEGHORN. 

-      $2,300 

-  1.05 

$5.30 

-  1.591 
2.490 

.800 

$2,500 

5.044 

.800 

$2,511 
15.811 

.800 

$2,310 
.770 

$1,855 
1.000 

$9,278 


Louis,  -  -  .  .  4.852 
Demi  Louis,  -  -  2.336 
Spanish  dollars,      -        -     1.000 

AUSTRIAN    DOMINIONS. 


Souverein,    - 
Double  ducat,  - 
Hungarian  do. 
Rix  dollar,  s.    - 

BAVARIA. 

Carolin, 


-  $3,377 

4.589 

2.996 

.960 

-  $4,957 


Ducat,  p. 


Pistole, 


BERNE. 


BRUNSWICK. 


COLOGNE. 


Ducat,        -        -        - 

EAST    INDIES. 

Rupee,  Bombay,  (1818) 
Rupee,  Madras,  (1818) 
Pagoda  Star, 
Sicca  Rupee,  (Calcutta)  s 


$1,986 

$4,548 

$2,667 

$7,096 
7.110 
1.798 
.480 
.450 


Rupee,  Bombay,  Madras,  s. 

WEST    INDIES. 

Spanish  gold  and  silver  coins. 

FRANKFORT    ON    THE    MAINE. 

Ducat,       -        -        -        $2,799 

SWITZERLAND. 

Pistol  Hel'c  Repub.  (1800)  $4,560 


REDUCTION    OF    FOREIGN    CURRENCIES. 

10.  To  change  English  or  Sterling  money  to  American  or  Federal 
money,  and  the  reverse  : — 

RULE. 

11.  Reduce  the  given  sum  to  the  decimal  of  a  pound,  then  multiply 
it  by  40  and  divide  hy  9,  to  produce  Federal  money :  and  reverse  the 
process  to  produce  sterling  money  again. 

12.  Reduce  £370.  6s.  3d  sterling  to  Federal  money. 

13.  Reduce  $1645.83333+ to  sterling  money. 

14.  Reduce  jC457.  17s.  6d.  sterling  to  Federal  money. 

15.  Reduce  $2,035  to  sterling  money. 

16.  The  reduction  of  sterling  money  is  of  so  frequent  occurrence 
among  commercial  men,  that  it  is  very  desirable  to  have  the  process 
rendered  as  easy  and  as  concise  as  possible,  as  is  attempted  in  the 
following  rule. 

RULE. 

17.  Take  half  the  shillings  for  the  tentJis  of  a  pound,  and  if  the 
shillings  be  odd,  call  the  hundredths  5  ,*  next  reduce  the  pence  and 

Q.  What  IS  c.ie  value  of  the  Johannes  of  Portugal  ?— of  the  ducat  and  ruble 
of  Russia  ?— of  the  rix  dollar  of  Sweden  ? 


286  ARITHMETIC. 

farthings  to  farthings^  for  thousandths  and  hundredths  respectively, 
then  the  sum  of  the  whole  increased  by  1  thousandth  lohen  the  far- 
things are  above  12,  and  by  2  thousandths  when  they  are  above  36, 
will  express  the  decimal  required* 

18.  Reduce  jC815.  16s.        16s-=  .8      =^  of  16s. 
6|(1.  to  the  decimal  of  a  £.         e^d.  =  .02  5  =  Farthings  in  6l-d. 

A.  £815.826.  .00  1  for  the  excess  abovs  12. 

jG'  .  8  2  6 


19.  Reduce  £182.  9s.  |d.  to  the  decimal  of  a  £. 
A.  £182.  543. 


9s.  = 

■-£  .  45. 

3_ 

--    .003 

jE:.453 

A. 

JC240.669 

A. 

i:203.857 

A. 

£814.146 

A. 

JC317.002 

A. 

iE:215.054 

A. 

$.955,795^ 

20.  Reduce  £240.  13s.  4Ad.  to  the  decimal  of  a  jG 

21.  Reduce  £203.  17s.  l^d.  to  the  decimal  of  a  £. 

22.  Reduce  JC814.  2s.  Ud.  to  the  decimal  of  a  £. 

23.  Reduce  £317.  Os.  |d.  to  the  decimal  of  a  £. 

24.  Reduce  £215.  Is.  Id.  to  the  decimal  of  a  £. 

25.  Reduce  £215.  Is.  Id.  to  Federal  money. 

26.  Reduce  $62,354.62^  to  sterling  money.  A.  i:i4,029.  15s.  9^d. 

27.  To  reduce  the  denomination  of  any  foreign  currency  to  Fede- 
ral money : — Multiply  by  its  value  in  Federal  money  found  in  the 
table. 

28.  Reduce  500  reals  plate  to  Federal  money. 

29.  Reduce  50  dollars  Federal  money,  to  reals  plate. 

30.  Reduce  250  guilders  to  Federal  money. 

31.  Reduce  100  dollars,  Federal  money,  to  florins. 

32.  Reduce  37  francs  and  5  decimes  to  ^.  A.  $6,975. 

33.  Reduce  215  roubles  and  5  grievfener  to  $.  A.  $161.62}. 

34.  Reduce  417  rix  dollars,  (Prussia)  to  Fed.  money  A.  $285.41^. 

35.  Reduce  240  skillings,  (Sweden)  to  Fed.  money.  A.  ^5.25 

36.  Reduce  415  marks,  (Denmark)  to  Fed.  money.        ^4..  36.65f. 

37.  Reduce  $4.50  to  pezzas,  (Genoa.)  A.  5  pezzas. 

38.  Reduce  $8.50  to  paoli,  (Rome.)  A.  85  paoli. 

39.  Reduce  200  ducati  and  6  carlini  to  $.  A.  160.48. 

40.  Reduce  700  tari,  (Malta,)  to  Federal  money,  4.  $23.33.^. 

41.  Reduce  1  oncia  and  15  tari  to  S.  tI.  $3.60. 

42.  Reduce  1100  rupees  5  annas  to  Federal  money.  A.  S610.77f. 

Q.  What  is  the  rule  for  changing  sterling  money  to  Federal  money  and  the 
reverse  ?  11.  "What  is  a  more  concise  rule  ?  17.  How  is  the  denomination  of 
any  foreign  cunency  reduced  to  Federal  money,  and  the  reverse  ? 

*  Since  shilUngs  are  twentieths  of  a  pound,  half  their  number  must  be  tenths,  and  if 
there  be  a  remainder  it  is  of  course  .1=  .10,  the  half  of  which  is  .05  or  five  hundredths. 
Again,  £l  =  960qr.  or  960ths  of  a  pound,  but  if  1,000  farthings  instead  of  960  were  equal 
to  \£,,  any  number  of  farthings  would  make  the  same  number  of  thousandths.  But  960 
increased  by  l-24lh  part  of  itself  is  1 ,000 ;  if  therefore,  we  increase  any  number  of  far- 
things by  the  2'lth  part  of  itself,  the  sum  will  be  an  exact  decimal;  wherefore,  if  tho 
number  of  farthings  exceed  12,  the  l-24th  part  of  them  is  greater  than  ^qr.  and  there- 
fbre  1  must  be  added,  and  when  their  number  exceeds  36,  ]-24lh  part  is  greater  than  1  ^l'- 
and  therefore  2  must  be  added.  The  I  and  2  are  properly  1  and  2  thousandths,  because 
they  occupy  those  places  in  the  decimal. 


REDUCTION    OF    FOREIGN    CURRENCIES.  287 

43.  Reduce  $303  to  guineas,  (England.)     See  Table.  A.  40. 

44.  Reduce  40  guineas  to  American  eagles,  (new.)      A.  20E.  $3 

45.  Reduce  450|-  sovereigns  to  Federal  money.       A.  $2,183,123. 

46.  Reduce  100  shilling  pieces,  (England,)  to  dollars.  A.  $22. 

47.  Reduce  40  double  louis  (old)  to  English  crowns.        A.  352f  |. 

48.  Reduce  80  Napoleons  to  American  eagles.  A.  30E.  $8.08cts. 

49.  Reduce  100  Spanish  coronilla  to  sovereigns.  A.  20S.  5s.  8|d. 

50.  Reduce  17,000  Johannes,  (Port.)  to  Napoleons,    ^.75,327.+ 

51.  Reduce  2,523  gold  Lions,  (Belgium,)  to  $.      A.  $12,731,058. 
62.  Reduce  1,333^  ducats,  (new,  Russia,)  to  $.     A,  $3,063.049|r 

53.  Reduce  8,042'Chr.  d'ors  to  ducats,  (Sicily.)      A.  40,421yV+- 

54.  Reduce  10,668  due.  (Cologne,)  to  pistoles,  (Bk.)  A.  6,255pf 

55.  Reduce  3,500  silver  rupees,  (Madras,  E.  Indies,)  to  CaroHns, 
(Bavaria.)  ^.317^  nearly. 

56.  How  many  dollars,  in  Federal  money,  will  pay  a  debt  of  £450 
•3s.  6d.  in  Great  Britain  ]  A.  $2,000,777^. 

57.  How  many  reals  plate,  in  Spain,  may  be  purchased  for  £234 
sterling?  ^.  10,400  reals  plate. 

58.  In  cases  like  the  last,  either  proceed  as  in  the  Rule  of  Three, 
or  first  reduce  to  Federal  money. 

59.  How  many  rix  dollars,  (Prussia,)  will  discharge  a  debt  of  1800 
francs,  (France  1)  A.  489^1  rix  dollars. 

60.  A  bought  400  yards  of  silk,  in  Paris,  at  3  francs  per  yard. 
What  was  its  cost  in  Federal  money  \  A.  $223.20. 

61.  How  many  rigsbank  dollars,  (Denmark,)  will  purchase  1,575 
liras  in  Genoa?  A.  468^^. 

62.  How  many  American  dollars  will  cancel  a  debt  of  7,400  tales 
in  China?  A.  $10,952. 

63.  United  States  on  England,  (cxv.  31.)  Reduce  £840.  14s.  6d. 
sterling,  to  Federal  money,  exchange  at  3  per  cent,  advance. 

A.  $3,848.652|. 

64.  Great  Britain  on  the  United  States.  What  sum,  in  London, 
will  purchase  a  Bill  on  Boston  for  $8,967.75,  exchange  at  4  per  ct 
discount?  A.  i:i937.  8d.+ 

65.  What  sum  in  American  eagles,  (old,)  v/ill  purchase  4,500 
guineas,  at  a  premium  of  3  per  ct.?  (cxv.  33.)  A.  2,027E.  $5.59.+ 

66.  United  States  on  France.  Reduce  1,174  francs  60  centimes 
to  Federal  money,  exchange  at  5  francs  40  centimes  per  dollar. 

A.  $217,519  nearly. 

67.  France  on  the  United  States.  Reduce  $15,000,000,  the  cost 
of  the  Louisiana  territory,  to  the  currency  of  France,  exchange  at  5 
francs  39  centimes  per  dollar.  A.  80,850,000  francs. 

68.  United  States  on  Amsterdam.  Reduce  1,250  florins  to  Fede- 
ral money,  exchange  39  cents  per  florin.  A.  $487.50. 

69.  Russia  on  the  United  States.  Reduce  240  rubles  5  grieveners 
to  Federal  money.     Exchange  at  par.  A.  $180.37^. 


ARITHMETIC. 

70.  United  States  on  Denmark.  Reduce  424  rigsbank  dollars  to 
Federal  money,  exchange  at  2  per  cent,  in  favor  of  Denmark. 

A.  S229.214yV 

71.  A  merchant  in  Boston  bought  in  London  40  pieces  of  black 
broadcloth,  each  containing  29y  yards,  for  13s.  6d.  per  yard.  How 
many  eagles  (new)  at  a  premium  of  2  per  cent.,  must  he  remit  to 
settle  the  bill  ^  ^.350  eagles. 

72.  If  you  purchase  in  China,  20,0001b.  of  tea,  for  2  maces  per  lb. 
which  you  sell  in  Amsterdam  for  1  guilder  per  pound,  worth  in  the 
United  States  2  per  cent,  advance,  and  with  the  proceeds  purchase  a 
bill  on  New  York  at  5  per  cent,  discount,  what  will  be  your  profit  in 
the  whole  transaction  ]  A,  $2,669.473|f . 


MISCELLANEOUS   EXAMPLES. 

CXVII.  1.  Suppose  you  employ  a  capital  of  $3,000,  and  invest 
315  dollars  9  cents  in  cloths,  176  dollars  6|  cents  in  linens,  3|  times 
the  last  amount  in  silks,  518  dollars  8  dimes  5  mills  in  various  other 
foreign  articles,  and  the  balance  in  domestic  cottons  at  12|  cents  per 
yd.;  how  many  yds.  could  you  purchase'?  A.  10,990yd,  2qr.  l^}na. 

2.  TSuppose  a  person  at  Boston,  and  another  at  Philadelphia,  the 
distance  between  each  place  being  about  300  miles,  set  out  to  meet 
each  other  on  the  road.  Required  how  far  they  are  distant  when 
each  has  traveled  79m.  5fur.  200  yd.'?     A.  140m.  4fur.  7rd.  Uyd. 

3.  The  sum  of  £1,261  was  left  a  person  in  the  United  States,  by 
a  relative  in  England,  to  be  divided  between  a  mother,  son,  and 
daughter,  so  that  the  son  shall  have  three  times  as  much  as  his 
mother,  and  the  mother  double  that  of  the  daughter ;  required  the 
share  of  each. 

A.  The  daughter's,  £UOh  the  mother's,  je280f ;  the  son's,  jC840f . 

4.  A  tract  of  land  measuring  7,495A.  3R.  32rd.  is  to  be  divided 
among  a  regiment  consisting  of  a  colonel,  a  major,  5  captains,  9  lieu- 
tenants, 6  ensigns,  20  sergeants,  and  450  privates,  so  that  a  sergeant 
is  to  have  twice  as  much  as  a  private,  an  ensign  8  shares,  a  lieutenant 
12,  a  captain  20,  the  major  30,  and  the  colonel  50 ;  required  the  share 
of  each,  and  the  value  of  the  land  at  3  dollars  per  acre  1 

Answers. — Soldiers,  9A.  12rd.=  $27.22i;  sergeants,  18A.  24rd. 
=  $54.45;  ensigns,  72A.  2R.  16rd  =$217.80 ;  lieutenants,  108 A. 
3R.  24rd.=  $326.70;  captains,  181A.  2R.=  $544.50;  major,  272A. 
1R.=  $816.75;  colonel,  453A.  3R.=  $1,361.25. 

5.  Jacob,  by  contract,  was  to  serve  Laban,  for  his  two  daughters, 
14  years;  when  he  had  accomplished  lOY.  lOmo.  lOwk.  lOda.  lOh. 
10m.,  how  many  minutes  had  he  then  to  serve  1        A.  1,416,350. 

6.  Suppose  you  were  15  years  old  on  the  first  day  of  January,  1810, 
how  many  seconds  old  must  you  have  been  on  the  first  day  of  August, 
1840,  making  due  calculations  for  the  leap  years  and  the  days  in  each 
month  from  one  date  to  the  other  ?         A.  1,438,473,600  seconds. 


MISCELLANEOUS    EXAMPLES.  289 

7.  How  much  time  has  elapsed  from  Nov.  19th,  1826,  to  Jan.  15th, 
1830 ;  from  April  19th,  1815,  to  Sept.  20th,  1837 ;  and  from  June 
12th,  1836,  to  March  1st,  1839 1 

A.  3y.  Im.  26d.;  22y.  5m.  Id.;  2y.  8m.  19d. 

8.  How  many  barrels  can  be  filled  with  60  hogsheads  of  molasses, 
each  containing  62  gallons,  3  quarts,  1  pint,  3  gills'?     A.  119|fbl. 

9.  A  vintner  bought  138  gallons  of  wine,  at  10s.  a  gallon,  of  which 
he  retained  18  gallons  for  his  own  use  ;  at  what  rate  must  he  sell  the 
remainder,  that  he  may  have  his  own  for  nothing"?         A.  $1.91f. 

10.  The  national  debt  of  England,  some  time  ago,  amounted  to  820 
millions  sterling. 

11.  Required  the  number  of  shillings  and  guineas  (21s.)  in  that 
sum.  A.  16,400,000,000s.;  780,952,381  guineas.+ 

12.  Required  the  weight  of  the  debt  in  guineas,  (gold,)  each  weigh- 
ing 5dwt.  9gr.  A.  17,490,0791b.  4oz.  7dwt.  21gr. 

13.  Required  the  weight  of  the  debt  in  shilling  pieces,  (silver,) 
each  weighing  3dwt.  21gr.  A.  264,79 l,666f lb. 

14.  Required  the  weight  in  one  pound  notes,  (£1,)  120  weighing  1 
ounce.  '  A.  213T.  lOcwt.  3qr.  81b.  5^oz. 

15.  Required  the  time  it  would  take  a  person,  allowing  him  to 
count  100  pieces  or  notes  in  a  minute  for  10  hours  of  the  day,  (Sun- 
days excepted,) — 

To  count  the  debt  in  guineas.     A.  41Y.  182da.  8h.  44m.  nearly. 
To  count  the  debt  in  shillings.  A.  873Y.  84da.  3h.  20m. 

To  count  the  debt  in  notes,  (£1.)  A.  43Y.  207da.  6h.  40m. 

16.  Required  how  many  wagons,  loaded  with  1,2001b.  each,  would 
be  sufficient — 

To  carry  the  debt  in  shilling  pieces.  A.  220,660,  nearly. 

To  carry  the  debt  in  gold  (guineas).  A-  14,576,  nearly. 

To  carry  the  debt  in  pound  notes.  A.  356,  nearly. 

17.  Required  the  number  of  miles  the  wagons  would  extend,  allow- 
ing 30  feet  to  each  wagon  and  horses — 

When  they  are  loaded  with  shillings.  A.  1,253m.  6fur. 

When  they  are  loaded  with  guineas.  A.  82m.  6|fur. 

When  they  are  loaded  with  pound  notes.  A.  2m.  7rd.  l^yd. 

18.  Required  the  yearly  interest  of  the  debt.  J..  ^218,666,666.66f. 

19.  Required  the  number  of  years  that  would  be  required  to  pay 
off  the  debt  by  an  annual  assessment  of  9  pence  per  pound  on  the 
value  of  the  whole  property  in  Britain,  which,  according  to  the  esti- 
mate of  Dr.  Colquhoun,  is  jC2,736,640,000.       A.  Almost  8  years. 

20.  Required  what  per  cent,  and  what  sum  on  the  pound  the  Bri- 
tish proprietors  must  assess  themselves,  supposing  that,  by  a  generous 
exertion,  they  agree  to  pay  off  the  debt  at  once. 

A.  30  per  cent,  nearly. 

21.  What  is  the  greatest  common  divisor  of  204, 1,190, 1,445,  and 
2,006]  ^.17. 

25 


390  ARITHMETIC. 

22.  Find  the  greatest  common  measure  of  63  and  168,  with  which 
divide  their  sum,  difference,  and  common  multiple.  -4.21. 

23.  Hence,  if  any  number  will  measure  each  of  two  others,  it  will 
measure  their  sum,  their  difference,  and  their  common  multiple. 

24.  Since  the  common  measure  of  two  or  more  numbers  becomes, 
when  those  numbers  are  multiplied  together,  or  when  each  is  multi- 
plied into  itself,  a  factor  in  every  product,  therefore — 

25.  A  common  measure  of  two  or  more  numbers  will  measure  the 
squares,  cubes,  6fC.  of  those  numbers,  also  the  continued  product  of 
those  same  numbers  into  each  other. 

26.  What  is  the  least  common  multiple  of  2,  3,  4,  6,  7, 12,  and  14] 

A.  84. 

27.  Reduce  24,  6|,  12^,  and  54x^5  to  their  equivalent  improper 
fractions.  A.  V ;  \' ;  W  ;  Vt  . 

28.  Reduce  to  improper  fractions  41t\,  123t\,  275||,  and  374yV5- 

A      540  .  2095  .  4139  .  3J-516 
-^'  T3"  »  ~TT~  >  ~\'S~t        \0T^- 

29.  Reduce  to  mixed  numbers  13f  of  7^^,  f  of  ^  of  12|,  and  15-|^ 
of  8f  of  13f .  A.  99H ;  4i ;  1,847^. 

30.  Reduce  50  to  a  simple  fraction.  A.  -A?r. 

^^ 

132 

31.  Reduce  to  mixed  numbers  «^V ;  ^^ ;  ^  of  4  of  6| ;  |  of  -^• 

A.  SO^VV;  52if|;  1t\;  2ff.' 

32.  Reduce  to  prime  terms  aV^j;  ii§|f;  ^mVoU^  M^- 

A       2  .      50    .        7.11 
-^-   7  J   TFT  >   UTS  }   Tff* 

33.  Find  the  least  common  denominator  of  4,  §,  f ,  §>  and  |. 

A        30".     40  .     45  .     48  .     5  0 

-^'  wb  i  Fo"  >  "STi  J  ¥0"  J  Tnr* 
71 

34.  Find  the  least  common  denominator  of  2^,  f  of  9f ,  and  gg 

A        2075  .     3196  .     3150 
^'   TT?ir>   TTSTTj   TTSIT' 

35.  What  is  the  value  of  1^+|  of  4j+|-r  1  A.  5^. 

Al 

36.  What  is  the  difference  between  f  of  rf  and  f  of  f|  1— between 

If  and  ^]  ^.  M;i^' 

37  How  many  times  is  the  sum  of  5^  and  3|  greater  than  their 
difference  1  A.  4:  times. 

2r  4l 

38.  Multiply  together  -^  of  ^,  and  |  of  7^.  A.  ^^■^. 

39.  Find  the  continued  product  of  f|f,f^|^,^^,  and  mi.  A.  1. 

40.  Find  the  value  of  |  of  ^-^j  of  3Hf  of  3|.  A.  Iff. 

41.  Express  ^^  of  a  yard  as  the  fraction  of  an  inch,  and  IH  o^  an 
inch  as  that  of  a  pole.  _4.  |4.  _^^ 


MISCELLANEOUS    EXAMPLES.  291 

42.  Required  the  sum  and  difference  of  f  of  a  pound,  and  f  of  a 
guinea,  (21s.)  A.  £1.  2s.  8d.;  4s. 

43.  Reduce  ^r  of  a-  pound,  -^  of  a  guinea,  and  I  of  3s.  9^d.  to 
fractions  of  the  same  denomination,  and  to  the  same  denominator. 

A  7040  .  -rose  .  7007 
•  73"?1"»  ■^asri'j  -ys^a- 

44.  The  aggregate  of  f  of  f  of  a  sum  of  money  is  $133  ;  what  is 
that  sum  1  A.  $332|. 

45.  Find  the  fraction  which,  when  multiplied  by  f  of  f  of  3^,  gives 
a  result  equal  to  ^.  A.  y%. 

46.  A  person  having  f  of  a  coal  mine,  sells  f  of  his  share  for 
$2,000  ;  what  is  the  whole  mine  worth  ?  A.  $6,666. 66f. 

47.  The  third  part  of  an  army  was  killed,  the  fourth  part  taken 
prisoners,  and  1,000  fled.  How  many  were  there  in  the  armyl — 
how  many  killed  1 — how  many  taken  prisoners  1 

A.  2,400;  800  killed;  600  prisoners. 

48.  A  can  do  a  job  of  work  in  5  days,  B  in  6,  and  C  in  7 ;  how 
much  can  they  jointly  do  in  2  days  1  A.   lyf  j  job. 

49.  The  owner  of  yV  of  a  ship  sold  y^r  off  of  his  share  for  $12^; 

2- 
what  would  jf  of  ?  cost  at  the  same  rate  1  A.  $200. 

50.  If  a  cask  be  emptied  by  two  taps  in  4  and  6  hours  respectively, 
in  what  time  will  it  be  emptied  by  both  of  them  together,  the  rates  of 
•filux  remaining  the  same  throughout]  A.  2h.  24m. 

51.  A,  B  and  C  can  perform  a  piece  of  work  in  12  hours ;  also, 
A  and  B  can  do  it  in  16  hours,  and  A  and  C  in  18  hours  ;  what  part 
of  the  work  can  B  and  C  do  in  9^  hours,  and  in  what  time  would  A 
do  the  whole  ?  A.  | :  28h.  48ra. 

52.  What  decimal  fractions  are  equivalent  to  ys,  j^j,  and  ^ila  "^ 

^.  .16;  .072;  .006640625. 

53.  From  unity  take  .123456789.  A.  .876,543,211. 

54.  Find  the  continued  product  of  .275,  2.75,  and  27.5. 

A.  20.796875. 

55.  Find  the  sum  of  the  quotients  of  1.68  by  .024,  of  971.7  by 
123,  and  of  142.025  by  .0437,  and  prove  the  results  by  vulgar  frac- 
tions. A.  3,327.9. 

56.  One  man  owns  .6  of  a  bank,  another  |,  and  Mr.  Darby  the  rest ; 
what  is  Darby's  part,  and  what  the  value  of  each  part,  allowing  the 
capital  of  the  bank  to  be  $100,000 1 

A.  $27,500;  $60,000;  $12,500. 

57.  In  a  certain  school,  .125  of  the  pupils  study  geography,  .3  study 
grammar,  |  arithmetic,  and  18  learn  to  read.  What  is  the  number  in 
each  branch  1  A.  Geography,  30  ;  grammar,  72  ;  arithmetic,  120  ; 
reading,  18.     Whole  number,  240. 

58.  A  man  whose  annual  income  is  $3,000,  spends  .12  of  it ;  how 
many  dollars  will  he  have  saved  at  the  end  of  each  year  1 

A.  $2,640. 


292  ARITHMETIC. 

59.  If  a  man  in  trading  adds  annually  to  his  capital  20  per  cent., 
how  many  years  will  be  required  for  his  capital  to  double. 

A.  5  years. 

60.  What  will  be  the  value  of  8cwt.  3qr.  141b.  of  sugar  at  $5}  per 
cwt.?  A.  $46.67|. 

61.  What  is  the  amount  of  $300  from  Jan.  1st,  1830,  to  Nov.  19th, 
1834?  A.  $387.90. 

62.  How  much  is  the  present  worth  of  $1,000,  discounted  at  bank, 
and  payable  in  4  months  (which  includes  the  3  days  grace)  less  than 
the  present  worth  of  the  same  sum  for  the  same  time,  calculated  as  in 
Lxxxiii.?  A.  89c.  2/ym. 

63.  Suppose  a  minor,  now  15  years  old,  is  to  receive  a  legacy,  at 
21,  of  $27,200  ;  what  is  its  present  value  1  A.  $20,000. 

64.  When  a  merchant  wants  a  loan  at  bank  of  $3,958,  for  60  days, 
what  sum  must  be  specified  in  the  note  to  receive  that  sum  1 

A.  $4,000. 

65.  What  sum  in  cash,  reckoning  compound  interest,  is  equivalent 
to  $3,207.13^05  due  20  years  hence,  without  interest  1  A.  $1,000. 

66.  A  merchant  bought  cotton  goods  to  the  amount  of  $1,300  for 
cash ;  he  kept  them  on  hand  1  year  and  8  months,  then  sold  them  for 
15  per  cent,  advance  on  their  cost,  but  on  4  months'  credit ;  what 
was  his  profit  ?  A.  $35,685+. 

67.  Suppose  a  bookseller  purchases,  at  "  trade  sale,"  books  to  tl^ 
amount  of  $500,  the  terms  being  indisputable  paper  for  6  months,  or 
4  per  cent,  discount  for  cash,  and  he  chooses  the  latter,  what  does  he 
make  by  advancing  the  cash  1  Discount  made  at  the  time  of  the  pur- 
chase is  usually  computed  like  bank  discount.  A.  ^5.436. 

68.  A  merchant  buys  on  credit  goods  amounting  to  $2,060,  but  for 
cash  down  gets  a  deduction  of  5  per  cent,  bank  discount ;  keeps  them 
on  hand  60  days,  then  sells  them  on  one  year's  credit,  at  20  per  cent, 
advance  from  the  purchase  price.  What  is  the  present  worth  of  his 
clear  profit  at  the  time  of  sale "?  A.  $355.50. 

69.  The  bill  for  the  union  of  the  Canadas,  in  1840,  provided  that 
the  governor  general  should  receive  an  annual  salary  of  £7,000. 
How-Tiuch  does  that  exceed  the  salary  of  the  president  of  the  United 
States,  which  is  $25,0001  A.  $6,1 11.1  Hi 

70.  To  find  the  equated  time  for  the  payment  of  bills  due  at  differ- 
ent times. — Multiply  each  sum  by  the  time  it  has  to  run,  {computing 
from  the  date  of  the  first,)  and  divide  the  sum  of  the  products  by  the 
amount  of  the  debt. 

71.  An  agent  sells  goods  for  his  employer,  payable  at  the  following 
dates— $120  due  7th  of  January  ;  $130  due  9th  of  February ;  $200 
due  15th  of  March ;  $500  due  20th  of  May ;  $240  due  15th  of  Au- 
gust.    Required  the  average  time  of  payment.  A.  May  2d. 

72.  The  prices  of  goods  are  often  named  in  pounds,  shillings,  &c., 
while  their  amount  is  carried  out  in  federal  money,  as  follows : — 


MISCELLANEOUS    EXAMPLES.  293 

Messrs.  Bates  &  Allen,  Bought  of  Brown  &  Dobson, 

Wheat,  1,000  bushels,  at  7s.  6d        -        -        -        $ 
Salt,       1,300  bushels,  at  3s.      - 
Rye,      2,500  bushels,  at  4s.  6d.        -        -        - 

Oats,      1,800  bushels,  at  Ss.  3d.        -        -        -        

$4,450. 

73.  The  notes  referred  to  below  are  to  be  calculated  according  to 
the  rules  under  which  they  are  given  as  examples.  In  these  opera- 
tions, the  pupil  will  derive  much  aid  from  the  Table  in  Compound 
Interest. 

FOR   RULES   SEE   LXXXI. 

74.  What  is  the  balance  due  at  compound  interest  on  Note  25  ? 
Results— 357,304+;  115,730;  56,047;  12,240.     A.  $63,127+. 

75.  What  is  the  balance  due  at  compound  interest  on  Note  26  % 
Results— 596,996  +  ;  22,472;  43,778;  248,676.     A.  $79,822  +  . 

76.  What  is  the  balance  due  at  compound  interest  on  Note  33? 
Results— 2,472  ;  236,856  ;  210,476.  A.  $1,300,458,  nearly. 

77.  What  is  the  balance  due  at  compound  interest  on  Note  35  ? 
Results— 64,236 ;  472,418;  433,641;  337,228.    A.  $197,057+. 

78.  What  is  the  balance  due  at  compound  interest  on  Note  39  ? 
Results— 81,265  ;  710,869;  662,824;  464,715.       A.  $479,423. 

79.  Find  a  4th  proportional  to  35 :  aV  -  ^f  5  also  to  125  :  .0145 : :  35. 

A.  j^  and  .00406. 

80.  If  two  men,  A  and  B,  together,  can  finish  a  piece  of  work  in 

10  days,  and  A,  by  himself,  in  18  days,  what  time  will  it  take  B  to 
do  the  whole  1  A.  22|  days. 

81.  Three  agents,  A,  B,  and  C,  can  produce  a  given  effect  in  12 
hours ;  also,  A  and  B  can  produce  it  in  16  hours,  and  A  and  C  in  18 
hours  ;  in  what  time  can  each  of  them  produce  it  separately  ? 

A.  A,  28§h.;  B,  36h.;  C,  48h. 

82.  Distribute  $200  among  A,  B,  C,  and  D,  so  that  B  may  receive 
as  much  as  A;  C  as  much  as  A  and  B  together,  and  D  as  much  as  A, 
B,  and  C,  together.  ^.  A's,  $25;  B's,  $25;  C's,  $50;  D's,  $100. 

83.  At  what  time  between  2  and  3  o'clock  are  the  hour  and  minute 
hands  of  a  clock  together  1  At  two  o'clock  the  hour  hand  is  two  of 
the  portions  called  hours  of  one  hand,  and  five  minutes  of  the  other, 
in  advance  of  the  minute  hand  ;  and  their  rates  being  as  1  :  12,  the 
minute  hand  gains  55  in  60,  or -11  in  12,  upon  the  hour  hand ;  hence, 

11  :  2  :  :  12  :  2h.  10|fm.  A.  10||m.  past  2  o'clock. 

84.  A  person,  on  looking  at  his  watch,  was  asked  the  "  time  of 
day."  He  replied,  that  it  was  between  4  and  5,  and  the  minute  and 
hour  hand  were  together ;  what  was  the  exact  time  1 

A.  21m.  49yYsec.  past  4. 

85.  Two  clocks  point  out  12  at  the  same  instant ;  one  of  them  gains 
7sec.  and  the  other  loses  8sec.  in  12  hours  ;  after  what  interval  will 
one  have  gained  half  an  hour  of  the  other,  and  what  o'clock  will  each 
then  show?  A.  60  days ;  14m.  past  12 ;  16m.  of  12. 

25* 


294  ARITHMETIC, 

80.  A  father  divided  his  estate  among  his  three  sons,  giving  to  A 
SlO  as  often  as  to  B  $6,  and  to  C  but  S3  as  often  as  to  B  $7,  and 
yet  C's  dividend  was  $4,800.  What  did  the  whole  estate  amount  to  1 

A.  S34,666f. 

87.  If  1  dollar  or  6  shillings  in  New  England  be  equal  in  value  to 

8  shillings  in  New  York;  £1.  8s.  in  New  York  to  jCl.  Cs.  3d.  in 
New  Jersey;  £1.  7s.  6d.  in  New  Jersey  to  jCl.  9s.  4d.  in  North 
Carolina;  £l.  8s.  in  North  Carolina  to  17s.  6d.  in  Canada;  and 
£2.  10s.  in  Canada  to  £2.  5s.  sterling;  how  many  dollars  in  New 
England  are  equal  to  £450  sterling  ?  A.  S2,000. 

88.  A  person,  by  disposing  of  goods  for  $182,  loses  at  the  rate  of 

9  per  cent. ;  what  ought  they  to  have  been  sold  for  to  realize  a  profit 
of7percent.1  A.  $214. 

89.  A  stationer  sold  quills  at  lis.  a  thousand, by  which  he  cleared 
I  of  the  money,  and  he  afterwards  raised  them  to  13s.  6d.  a  thousand ; 
what  did  he  clear  per  cent,  by  the  latter  price  1 

A.  jC96.  7s.  3d.  lyVqr- 

90.  At  what  price  must  a  commodity,  purchased  at  the  rate  oi 
£14.  5s.  per  cwt.,  be  sold  to  gain  21  per  cent.,  and  how  much  must 
be  sold  to  clear  £100  ? 

A.  £17.  4s.  10]d.  per  cwt.;  33cwt.  Iqr.  16lb.  lluVrOz. 

91.  Divide  $64  among  A,  B,  and  C,  so  that  A  may  have  three 
times  as  much  as  B,  and  C  may  have  one  third  of  what  A  and  B  to- 
gether have  ?  A.  A's  $36  ;  B's  $12  ;  C's  $16. 

92.  A  person  paid  a  tax  of  10  per  cent,  on  his  income  ;  what  must 
his  income  have  been,  when,  after  he  had  paid  the  tax,  there  was 
$1,250  remaining  1  A.  $l,388.888f. 

93.  A  hare  starts  40  yards  before  a  greyhound,  and  is  not  per- 
ceived by  him  till  she  has  been  up  40  seconds  ;  she  scuds  away  at 
the  rate  of  10  miles  an  hour,  and  the  dog  pursues  her  at  the  rate  of 
18  miles  an  hour  ;  how  long  will  the  course  last,  and  what  distance 
will  the  hare  have  run  1  A.  QO^^sec,  490yd. 

94.  If  5  men  or  7  women  can  perform  a  piece  of  work  in  35  days, 
in  what  time  can  7  men  and  5  women  do  the  same  1  A.  lO^y  days. 

95.  If  15  men,  12  women,  and  9  boys,  can  complete  a  piece  of 
work  in  50  days,  what  time  would  9  men,  15  women,  and  18  boys, 
take  to  do  twice  as  much,  the  parts  done  by  each  in  the  same  time 
being  as  the  numbers  3,  2,  and  1 1  A.   104  days. 

96.  If  A  by  himself  can  do  a  piece  of  work  in  5  days,  B  twice  as 
much  in  7  days,  and  C  four  times  as  much  in  1 1  days,  in  what  time 
can  A,  B,  and  C,  together,  do  3  times  the  said  work "? 

A.  3  days  12h.  46y2^V- 

97.  If  A  can  do  a  piece  of  work  by  himself  in  1  hour,  B  in  3  hours, 
C  in  5  hours,  and  D  in  7  hours,  in  what  time  can  they  do  three  times 
as  much,  all  working  together  1  A.   Ih.  47m.  23fysec. 

98.  If  27  men  can  do  a  piece  of  work  in  14  days,  working  10  hours 
in  a  day,  how  many  hours  a  day  must  24  boys  work,  in  order  to  com- 


MISCELLANEOUS    EXAMPLES.  295 

plete  the  same  in  45  days,  the  work  of  a  boy  being  half  that  of  a 
man?  A.  7  hours. 

99.  If  10  cannon,  which  fire  3  rounds  in  5  minutes,  kill  270  men 
in  1|  hours,  how  many  cannon,  which  fire  5  rounds  in  6  minutes, 
will  kill  500  men  in  1  hour  at  the  same  rate  1  A.  20  cannon. 

100.  A  and  B  can  do  a  piece  of  work  in  10  days,  A  and  C  in  IS 
days,  and  B  and  C  in  14  days ;  in  what  time  can  they  do  it  jointly 
and  separately"?  A.  Together,  t^si  days;  A  in  17^^  days  ;  B  in 
22|f  days  ;  and  C  in  3 6 if  days. 

101.  If  120  men,  in  3  days  of  12  hours  each,  can  dig  a  trench  30 
yards  long,  2  yds.  broad,  and  4  feet  deep,  how  many  men  would  be 
required  to  dig  a  trench  50  yards  long,  6  feet  deep,  and  1^  yards 
broad,  in  9  days  of  15  hours  eachi  A.  60  men. 

102.  A  watch,  which  is  10  minutes  too  fast  at  12  o'clock  on  Mon- 
day, gains  3m.  lOsec.  per  day  ;  what  will  be  the  time  by  the  watch  at 
a  quarter  past  10  in  the  morning  of  the  following  Saturday] 

A.  40m.  36/^scc.  past  10. 

103.  Required  the  cavity  of  a  well,  whose  surface  measures  9 
Square  yards  5  feet,  and  depth  28}  fathoms.  A.  544yds.  18ft. 

104.  How  many  square  yards  in  the  area  of  Solomon's  Temple, 
whose  dimensions  are  mentioned  in  1  Kings ^  chap,  vi.,  reckoning 
the  cubit  to  be  18  inches  ]  A.  300. 

105.  The  length  of  Noah's  Ark  was  three  hundred  cubits,  the 
breadth  50,  and  the  height  30  ;  how  many  cubic  yards  did  it  contain, 
and  how  many  horses  might  have  been  lodged  in  it,  allowing  10  yards 
to  each  horse  t  A.  5,625  horses. 

106.  The  hold  of  a  vessel  is  120  feet  lon|,  33  feet  broad,  and  6 
deep ;  how  many  bales  of  goods,  each  measuring,  at  an  average,  6 
feet  by  4,  and  3  feet  deep,  may  be  stowed  in  her,  leaving  a  gangway 
3  feet  broad  1  A.  300  bales. 

107.  Required  the  square  feet  in  7  oak  planks,  each  23}  feet  long, 
and  their  several  breadths  as  follows — 4  of  13|  inches  in  the  middle, 
1  of  14}  inches,  and  the  other  2  each  16  inches  at  the  broader  end, 
and  13f  inches  at  the  narrower ;  and  what  will  be  the  cost  of  the 
whole  in  federal  money,  at  2s.  4^d.  sterling  per  foot  1 

A."l89ft.  4in.  81^',  $99.96,  nearly. 

108.  How  many  solid  feet  in  a  hewn  stick  of  timber  30  feet  longj 
1ft.  6in.  square  at  one  end,  and  2ft.  square  at  the  other  1  A.  92}ft. 

109.  How  many  solid  feet  in  a  round  log  24  feet  long  and  68  inches 
in  girt?  A.  48^  s.  ft.  =48  s.  ft.  288  s.  in. 

110.  How  much  in  length  of  a  tree  40  inches  in  girt,  will  make  a 
solid  foof?  A.  ll-^s  inches. 

111.  Required  the  solid  content  ofan  irregular  beech  log,  the  larger 
end  of  which  is  6  feet  by  50  inches  in  girt,  and  smaller  end  5  feet  by 
35}  inches  in  girt,  allowing  \  inch  upon  the  quarter  girt  for  the  bark. 

A.  8ft.  5in.  2^/.+ 

112.  A  maltster  has  a  kiln  18  feet  square,  which  he  intends  to  take 


i296  ARlfttMETIC. 

down,  and  build  a  new  one  which  shall  be  24  feet  in  breadth,  and  to 
dry  3  times  as  much  as  the  old  one,  required  its  length.     A.  40|-ft. 

113.  Bought  2,688  yards  of  cambric  at  8s.  8d.  a  yard,  and  sold  ^ 
at  10s.  2d.  per  yard  ;  ^  at  10s.  llh,  and  the  remainder  at  lis.  4^d. 
per  yard ;  what  is  the  whole  gain,  and  what  the  gain  per  cent.  1 

A.  £304.  14s  8d.  ;  £26.  2s.  4id.+ 

114.  At  what  times  between  2  and  3  o'clock  are  the  hour  and 
minute  hands  together;  at  right  angles,  and  in  opposite  directions  T 

A.  Together  at  lOfym.  past  2  ;  at  right  angles,  27fVm.  past  2 ;  in 
opposite  directions,  43f^ym.  past  2. 

115.  Four  men  bought  a  grindstone  of  60  inches  diameter.  Now 
how  much  of  the  diameter  must  be  ground  offby  each  man,  one  grind- 
ing his  part  first,  then  another,  and  so  on,  that  each  may  have  an 
equal  share  of  the  stone,  no  allowance  being  made  for  the  eye  1 

A.  1st.,  8.04  in. ;  2nd,  9.534  in.  ;  3d,  12.426  in.  ;  4th,  30in. 
lie.  The  wheels  of  a  carriage  are  2\  yards  asunder,  and  the  inner 
wheel  describes  the  circumference  of  a  circle,  whose  radius  is  20yd.; 
find  the  difference  of  the  paths  of  the  two  wheels.  The  circumfe- 
rence of  the  inner  circle =3. 14159x40  ;  the  circumference  of  the 
outer  circle=3. 14159x45;  whence  their  difference  is  evidently= 
3. 14159  X  5=  15.70795  yards  nearly.  The  45  is  double  the  radius  of 
the  outer  circle=  20  +  20+21+2^=45.  A.  15yVyds.  nearly. 

117.  Suppose  a  gentleman  from  London  contracts  to  construct  a 
railroad  in  the  United  States,  to  be  415m.  5fur.  30rd.  long,  at  the 
tate  of  £2,115.  19s.  6d.  sterling  per  mile,  and  is  able  to  complete 
only  f  of  that  distance,  what  sum  in  federal  money  ought  he  to  receive 
at  the  rate  he  was  to  be  paid  for  the  whole  ?  A.  SI, 563,823. 079 If. 

118.  A  person  being  asked  the  time  of  the  day,  replied,  that  the 
day  is  12  hours  long,  and  the  sun  rises  at  6  o'clock.  Now  if  you  add 
■|  of  the  hours  that  have  elapsed  since  the  sun  rose,  to  ^  of  those 
which  must  elapse  before  the  sun  sets,  you  will  have  the  exact  time 
of  the  day. 

1st.  Suppose  it  was  2  o'clock,  then  8  hours  must  have  elapsed 
since  sunrise,  |  of  which,  added  to  |  of  4  hours  make  7  hours  ;  then  8 
—  7==^1,  1st  error.  Next  suppose  that  it  was  10  o'clock,  then  4h. 
have  elapsed,  &c.  2nd  error,  4.  A.  12m.  past  1  o'clock. 

119.  If  A,  B  and  C,  could  reap  a  field  in  18  days;  B,  C  and  D,  in 
20  days ;  C,  D  and  A  in  24  days  ;  and  D,  A  and  B  in  27  days ;  in  what 
time  would  it  be  reaped  by  them  all  together,  and  by  each  of  them 
separately  1 

Answers.  All  together,  in  16-i^  days;  A,  in  87f|days ;  B,  in  60| 
days ;  C,  in  41,fV  days ;  and  D,  in  1701^  days 


QUESTIONS    FOR  EXAMINATION.* 

ADDRESSED ^0   THE    LEARNER   AND  CANDIDATES   FOR  SCHOOL-KEEPING. 

TO   BE   ANSWERED   WITHOUT  THE   SLATE.* 

CXVIII.  Q.  1.  What  is  Quantity !  See  page  26.  How  is  it  esti- 
mated U  Illustrate  it  by  an  example.  How  are  different  quantities 
expressed^ 

Q.  2.  What  is  Number  1  3.  How  is  it  represented?  What 
are  Concrete  and  Abstract  Numbers  1  26.  Simple  and  Compound 
Numbers'?  27.     What  is  meant  by  denomination'? 

Q.  3.  What  is  Arithmetic  '?  27.  What  are  its  fundamental 
rules  1    Describe  the  different  methods  of  representing  numbers  1  28. 

Q.  4.  What  is  Numeration  ?  32,  Notation  and  its  rule "?  Re- 
peat Numeration  Table  iii.  What  is  the  greatest  number  that  can 
be  formed  by  thirty  5s  1 

Q.  5.  What  is  addition  1  37.  Its  rule '?  Why  do  you  carry  1  for 
every  10'?  36.     What  is  the  proof  and  the  reason  for  it  ■?  37. 

Q.  6.  What  is  Subtraction '?  41.  Its  terms?  Rule?  Why  do 
you  borrow  from  one  figure  and  pay  to  another  ?  40.  What  is  the 
proof  and  the  reason  for  it?  42. 

Q.  7.  What  is  Multiplication?  45.  Its  terms?  Which  terms 
are  called  factors  and  why  ?  When  may  Addition  be  performed  by 
Multiplication?  How  do  you  multiply  by  12  or  less? — by  13  or 
more? — by  a  composite  number?— by  10,  100,  &c.  ?— 3700  by  210? 

Q.  8.  What  is  Division  ?  52.  Its  terms?  What  rule  is  proved, 
and  what  one  is  performed,  by  Division?  What  is  Short  Division  ? 
53.  Rule  ?  Long  Division  ?  54.  Rule  57.  Rule  for  dividing  by 
10,  100,  &c.  ?  60.  Rule  for  dividing  888  by  800  in  the  most  concise 
manner? 

Q.  9.  How  is  the  subtrahend  found  when  the  minuend  and 
remainder  are  given?  How  may  the  multiplier  be  found  with  the 
multiplicand  and  product  ?  How  may  the  remainder  be  found  with 
the  dividend  and  quotient  ? 

Q.  10.  What  is  the  amount  of  35  and  7500  and  30000  and  4200000 
added  together  ?  When  the  minuend  is  400  and  the  remainder 
40,  what  is  the  subtrahend  ?  When  the  multiplier  is  8  and  the  pro- 
duct 4000,  what  is  the  multiplicand  I 

Q.  11.  What  are  the  results  of  34398  muhiplied  by  100  and  the 
same  divided  by  100  ?  When  the  quotient  is  50  and  the  dividend 
2504,  what  is  the  remainder  ? 

*  Fertile  benefit  of  tlie  Teacher  the  answer  to  each  example  is  given  in  the  key,  to 
gethor  with  reasons  for  the  more  difficult  operations. 

t  All  questions  without  numbers  annexed,  reler  to  the  page  indicated  by  the  last  pre 
ceding  number. 


298  ARITHMETIC. 

Q.  12.  Two  men  are  traveling  in  the  same  direction,  one  at  the 
rate  of  36  miles  a  day,  and  the  other  40  miles;  in  how  many  days 
will  the  latter  be  100  miles  forward  of  the  other  1 

Q.  13.  Suppose  a  fox,  which  is  120  rods  before  ^  greyhound, 
runs  at  the  rate  of  4  rods  in  2  seconds,  and  the  dog  13  mdiS  in  6  sec- 
onds. How  far  must  the  dog  run  to  catch  the  fox,  and  how  long  will 
he  be  in  doing  it  1 

Q.  14.  If  the  minuend  be  600  and  the  difference  between  the 
remainder  and  subtrahend  100,  what  are  the  last  two  terms  T 

Q.  15.  Two  men  having  met  on  a  journey,  found  that  they  had 
both  traveled  1,200  miles;  but  one  had  traveled  200  miles  more 
than  the  other.    What  was  the  distance  each  traveled  1 

Q.  16.  Suppose  four  boys  together  weigh  435  pounds,  and  that  it 
should  so  happen  that  three  of  them  go  in  the  same  notch,  but  the 
other  in  a  notch  15  pounds  higher  ;  what  w^ould  be  the  weight  of  each 
boy  1 

Q.  17.  Divide  600  dollars  so  that  A  may  have  50  dollars  more 
than  B,  and  C  100  more  than  B. 

Q.  18.  Said  Harry  to  Dick,  my  purse  and  money  are  worth  100 
dollars ;  but  the  money  is  worth  19  times  more  than  the  purse.  How 
much  money  was  in  the  purse  1 

Q.  19.  If  a  clerk,  whose  salary  for  4  years  amounted  to  2,000 
dollars,  had  received  50  dollars  advance  for  each  successive  year  af- 
ter the  first,  what  was  his  annual  salary  1 

Q.  20.  Suppose  that  C  resides  3  times  as  far  from  Boston  as  A, 
and  D  5  times  as  far  as  C,  and  that  to  meet  in  that  city  they  must  all 
travel  380  miles.     What  distance  from  Boston  does  each  reside  1 

Q.  21.  A  man  bought  a  horse,  saddle  and  bridle  for  318  dollars, 
and  paid  20  times  as  much  for  the  horse  as  for  the  saddle,  and  5 
times  less  for  the  bridle  than  for  the  saddle.  What  did  th^  bridle  and 
saddle  both  cost "? 

Q.  22.  Fifteen  years  ago  I  was  three  times  as  old  as  my  eldest 
son,  who  was  then  but  15,  but  am  now  only  twice  as  old.  What  are 
our  present  ages "? 

Q.  23.  A  company  at  a  tavern  spent  ^Z-fo,  and  each  of  them  had 
as  many  dimes  to  pay  as  there  were  persons  in  the  company ;  how 
many  persons  were  there  I 

Q.  24.  What  is  the  difference  between  twice  twenty-five  and 
twice  five  and  twenty  1 

Q.  25.  A  snail  in  going  up  a  May-pole  22  feet  high,  ascended  4] 
feet  every  day,  and  descended  every  night  2f  feet ;  how  long  would  it 
be  in  getting  to  the  top  of  the  pole  1 

26.  What  is  Federal  Money  ?  73.  What  is  the  rule  for  Addition 
of  Federal  Money?  75.  For  Subtraction  1  76.  For  Multiplication  ? 
77.     For  Division?  78. 

Q.  27.  Suppose  you  sell  4  bushels  of  oats  at  37|  a  bushel,  and 
pay  62|  cents  for  5  pounds  of  cheese,  and  lay  out  the  balance  in  pins 


QUESTIONS  FOR  EXAMINATION.  299 

at  6  J  cents  a  paper ;  how  many  papers  of  pins  will  you  have,  and  how 
much  will  the  cheese  cost  by  the  pound  1 

Q.  28.  If  you  purchase  29  yards  of  ribbon  at  6|  cents  per  yard, 
and  give  the  shop  keeper  a  five  dollar  bill,  how  much  change  must  he 
give  you?  What  will  2  barrels  of  pork  cost  at  12^  cents  per  pound  1 
— at  10  cents  per  pound  1 

Q.  29.  What  will  6,404  articles  of  any  thing  cost  at  6y  cents 
each  ?— at  12|  cents  1 — at  1  shilling  1— at  25  cents  T— at  50  cents  1— 
at  75  cents  ? 

Q.  30.  What  is  Reduction  1  74.  Reduction  descending  and  its 
rule  ■?  89.  Reduction  Ascending  and  its  rule  1  What  is  Compound 
Addition  and  its  rule  ?  97.  Compound  Subtraction  and  its  rule  ?  101. 
Compound  Multiplication  and  its  rule  1  104.  Compound  Division 
and  its  rule  1  107. 

Q.  31.  What  are  Fractions ?  110.  What  is  the  denominator? 
Numerator !  How  is  the  integer  found  from  having  its  fraction 
given?  What  is  the  integer  of  which  f  is  20  ?  What  is  the  value 
of  a  fraction?  111. 

Q.  32.  What  part  of  48  is  36  ^  What  is  the  rule  and  reason  for 
it?  112.  What  are  the  two  ways  for  multiplying  a  fraction,  and 
why?  113.    What  for  dividing  fractions,  and  the  reason ?  114. 

Q.  33.  What  is  the  product  of  yf  ^  multiplied  by  47  ?— aW  mul- 
tiplied by  20  ?— ^0  divided  by  16?— /g  divided  by  3  ? 

Q.  34.  What  is  meant  by  a  common  measure  or  a  common  di- 
visor? 116.  What  by  a  common  multiple?  118.  What  are  the 
rules  for  finding  both?  117,  120.  What  is  the  greatest  common 
divisor  of  48  and  24  ?  What  is  the  least  common  multiple  of  5 
and  20  ? 

Q.  35.  What  is  a  Vulgar  Fraction  ?  121.  Describe  the  different 
sorts  of  Vulgar  Fractions  ?  How  are  mixed  numbers  reduced  to  im- 
proper fractions  ?  125.  Compound  and  complex  fractions  to  single 
ones  ?  123,  127.  Rule,  and  the  reason  of  it  for  the  reduction  effrac- 
tions to  their  lowest  terms  ?  122.  For  finding  the  least  common  de- 
nominator ?  127. 

Q.  36.  What  fraction  of  a  dollar  is  equal  to  f  of  a  pound  ?  What 
is  the  rule  for  it  ?  131.  What  are  the  rules  for  Addition  of  Frac- 
tions ?  133.  For  Subtraction?  134.  For  Multiplication?  136. 
For  Division?  139. 

Q.  37.  A  man  being  asked  how  long  he  had  been  in  business, 
replied  that  the  time  he  spent  at  school  was  6  years,  and  that  f  of 
that  period  is  just  |  of  the  time  he  had  been  in  business  ;  what  was  a 
direct  answer  to  the  question  ? 

Q.  38.  4  of  21  is  f  of  what  number  ?  |  of  22  is  f  of  what  num- 
ber? 

Q.  39.  One  drover  said  to  another,  I  have  20  cows.  Well,  said 
the  other,  ^  of  §  of  your  drove  is  only  ^  of  mine.  How  many  cows 
had  both  ? 


300  ARITHMETIC. 

Q.     40.  I  of  24  is  f  of  how  many  times  8. 

Q.     41.  |- of  24  is  f  of  how  many  eighths  of  40  1 

Q.     42.  What  number  added  to  f  of  16,  will  make  f  of  60  ? 

Q.  43.  A  man  bought  2^-  bushels  of  rye  at  one  time,  and 
1|  bushels  at  another.  He  sold  3j  bushels  ;  how  much  had  he 
left? 

Q.  44.  A  man  bought  a  sheep  for  5]  dollars  and  a  calf  for  Ty^ij 
dollars  ;  what  did  he  give  for  both,  and  what  was  the  difference  in 
their  cost  1    What  fraction  added  to  f  and  ^  will  make  a  unit  1 

Q.  45.  There  is  a  mast  erected  so  that  ^  of  it  stands  in  the 
ground,  |  in  the  water,  and  28  feet  out  of  the  water  ;  how  long  is  the 
mast  1 

Q.  46.  Suppose  A  having  taken  a  job  of  work  does  ^  of  it  in  1 
day,  ^  of  it  the  second  day,  and  jj  the  third  day  ;  how  much  of  it 
does  he  do  in  the  three  days  1 

Q.  47.  Suppose  A  can  mow  a  certain  field  in  2  days,  and  B  in  4 
days ;  what  part  of  the  field  would  each  mow  in  one  day  1  What 
part  would  both  mow  in  one  day]  How  long  would  it  take  both  to- 
gether to  mow  the  field  1 

Q.  48.  Suppose  a  cistern  has  three  spouts,  and  that  one  will  fill  it 
in  2  hours,  another  in  3  hours,  and  another  in  4  hours  ;  in  how  many 
hours  would  it  be  filled  by  them  all  together  "? 

Q.  49.  Suppose  a  cistern  is  filled  by  two  spouts  in  4  and  12  minutes 
respectively,  and  is  emptied  by  a  tap  in  16  minutes;  how  many 
minutes  will  have  elapsed  before  it  is  filled,  when  they  are  all  left 
open  or  running,  the  influx  and  efflux  being  uniform  1 

Q.  50.  What  is  one  half  the  quarter  of?  What  part  of  three 
pence  is  a  third  part  of  two  pence  1 

Q.  51.  If  a  herring  and  a  half  cost  a  penny  and  a  half,  how  many 
may  be  had  for  11  pence  ?    What  number  is  that  of  which  9  is  f  ? 

Q.  52.  What  number  is  that,  the  3d  and  4th  parts  of  which,  taken 
together,  make  24^  ?     What  part  of  a  dollar  is  a  3d  part  of  a  cent  1 

Q.  53.  What  would  f  of  a  hogshead  of  molasses  cost  at  50  cents  a 
gallon  1 — I  of  a  cwt,  of  sugar  cost  at  10  cents  per  pound  1 

Q.  54.  What  part  of  an  eagle  is  a  fourth  part  of  a  dime  ?  How 
many  pecks  are  yg-  of  a  bushel  ? 

Q.  55.  The  aggregate  of  f  and  |  of  a  debt  is  60  dollars ;  what  is 
that  debt  ?    What  fraction  multiplied  by  f  of  f  of  3  will  make  f  ? 

Q.  56.  A  person  having  f  of  a  coal  mine,  sells  f  of  his  share  for 
600  dollars  ;  what  is  the  whole  mine  worth  1 

Q.  57.  A  can  do  a  piece  of  work  in  5  days,  B  in  3  days,  and  C  in 
10  days  ;  how  long  would  they  jointly  be  in  doing  it  1 

Q.  58.  If  ^  the  trees  in  an  orchard  bear  apples,  ^  pears,  ^  plums, 
40  of  them  peaches,  and  10  cherries,  what  number  of  trees  does  the 
orchard  contain  ?    What  is  that  number  of  which  ^+ j+^  is  45 1 

Q.  59.  When  6|  bushels  of  oats  cost  2f  dollars,  what  is  the  price 


QUESTIONS    FOR    EXAMINATION.  301 

by  the  bushel  ?    What  number  is  that,  to  which,  if  its  half  be  added, 
the  sum  will  be  12 "? 

Q.  60.  What  number  is  that,  to  which,  if  its  half,  its  third,  and  its 
fourth,  be  added,  the  sum  will  be  25 1  The  double  and  the  half  of  a 
certain  number,  increased  by  5  more,  wUl  make  26 ;  what  is  that 
number  1 

Q.  61.  If  a  horse  will  eat  8cwt.  of  hay  in  a  month,  a  cow  4cwt., 
and  a  calf  3c wt.,  in  what  time  will  they  all  consume  a  load  of  hay] 

Q.  62.  A  person  being  asked  how  much  money  he  had,  replied 
evasively  as  follows — If^  and  \  of  it,  with  18  dollars  more,  be  added 
to  it,  the  sum  would  be  4  times  as  much  as  he  had.  What  sum  had  he  1 

Q.  63.  What  number  is  that,  which,  if  increased  by  its  half,  its 
third,  and  2  more,  will  be  doubled  1 

Q.  64.  A  man  spent  one  third  of  his  life  in  England,  one  fourth  in 
Scotland,  and  the  remainder,  which  was  20  years,  in  the  United 
States.     To  what  age  did  he  live  ? 

Q.  65.  A  person,  after  paying  away  one  third  of  his  money,  together 
with  ten  dollars,  finds  that  he  has  remaining  15  dollars  more  than  its 
half;  what  sum  of  money  had  he  1 

Q.  66.  A  man  having  purchased  a  drove  of  cattle,  was  driving  them 
to  market,  when  he  was  met  by  a  gentleman,  who  inquired  of  him 
where  he  was  going  with  his  100  head  of  cattle.  Sir,  said  he,  I  have 
not  100,  but  if  I  had  as  many  more  as  I  now  have,  |  as  many  more, 
and  7|  head  of  cattle,  I  should  have  100.     How  many  had  he  ? 

Q.  67.  What  are  Decimal  Fractions]  145.  What  decimals  are 
equal  to  I"? — to  f? — to  |] — to  f  1 — ^What  is  the  rule  for  these  reduc- 
tions?  149. 

Q.  68.  What  decimal  of  a  pound  is  2s.  6d.1  What  is  the  value  of 
£125  ?     Rules  for  the  last  two  examples  1  152,  153. 

Q.  69.  What  is  the  general  method  of  proceeding  in  decimal  rules, 
and  why]  153.  Rulefor  Addition  of  Decimals'?  154.  Subtraction'? 
155.     Multiplication]  156.     Division!   158. 

Q.  70.  What  is  the  sum  of  .6  and  .03  and  .004  and  .0005  ]  How 
much  does  unity  exceed  .123456789  ] 

Q.  71.  Multiply  1.234  by  10  ;— by  100  ;— by  1,000. 

Q.  72.  Divide  123.4  by  10;— by  100 ;— by  1,000. 

Q.  73.  How  is  Reduction  of  Currencies  performed'?  161.  What 
number  of  shillings  make  a  dollar  in  the  different  states '?  What 
number  of  pence  is  equal  to  12|-  cents  in  these  same  states'? 

Q.  74.  What  is  Rate  per  Cent.?  163.  Rule  for  finding  the  per 
centage '?  When  5  yards  of  broadcloth,  that  cost  6  dollars  per  yard, 
sells  for  25  per  cent,  profit,  how  many  dollars  does  it  sell  for "? 

Q.  75.  What  price  must  be  put  on  molasses  that  cost  30  cents  a 
gallon,  to  gain  20  per  cent,  on  the  sale  of  it  ] 

Q.  76.  What  are  Stocks'?  166.     When  you  buy  stocks,  the  par 
value  of  which  is  $500,  for  10  per  cent,  advance,  and  sell  them  for 
15  per  cent,  discount,  what  is  your  loss  in  the  transaction'? 
26 


302  ARITHMETIC. 

Q.  77.  What  is  Commission'?  167.  What  is  your  commission  for 
selling  goods  amounting  to  $1,000,  on  f  of  which  you  are  to  have  3 
per  cent.,  and  on  the  balance  3  per  cent.? 

Q.  78.  What  is  Insurance  I  168.  Suppose  you  have  810,000  in- 
sured on  your  house,  and  $2,000  on  your  furniture  ;  what  will  your 
insurance  amount  to  at  the  rate  of  40  cents  on  $100  for  your  house, 
and  ^  per  cent,  for  your  furniture  ? 

Q.  79.  What  is  Interest]  171.  What  are  the  rules  for  calculating 
the  interest  for  years,  months,  and  days  ]  174.  What  is  the  interest 
of  $500  for  1  year  ] — for  2  years  ] — for  1  year  8  months  1 — for  2 
years  0  months  ] 

Q.  80.  What  is  the  amount  of  $600  for  2  months  1 — for  15  days  1 — 
for  20  days  1— for  25  days  1— for  27  days  ? 

Q.  81.  What  are  the  three  rules  for  casting  interest  on  notes? 
185,  186,  188.  Wherein  does  an  indorsed  note  differ  from  others? 
182.  What  is  Compound  Interest?  189.  Discount,  and  its  rule? 
192.     How  is  bank  discount  reckoned?   196. 

Q.  82.  When  you  offer  to  the  bank  a  note  of  S60,  payable  in  sixty 
days,  and  it  is  discounted,  what  sum  do  you  receive  ?  What  is  equa- 
tion of  payments  ?  197. 

Q.  83.  What  is  Proportion?  209.  Geomfitrical  Proportion  ?  210. 
Rule  of  Three  ?  204.  On  what  principle  is  the  Rule  of  Three  based  ? 
213.     How  is  the  Rule  of  Three  in  Fractions  performed  ?  207. 

Q.  84.  What  is  Compound  Proportion?  217.  Rule?  Conjoined 
Proportion?  220.  Rule?  Fellowship?  222.  Rule?  Compound 
Fellowship?  224.     Rule?  225. 

Q.  85.  When  brandy  is  ten  cents  a  gill,  what  is  it  a  pint  ?  What  is 
it  a  quart  \  What  is  it  a  gallon  ?  How  many  gallons  of  ale  at  ten 
cents  a  gallon  are  worth  two  gallons  of  brandy  at  ten  cents  a  pint  ? 

Q.  86.  If  60  bushels  of  oats  and  5  tons  of  hay  will  keep  2  horses  6 
months,  what  quantity  of  each  will  keep  4  horses  a  year  ? 

Q.  87.  A  merchant  compounded  with  his  creditors  for  75  cents  on 
the  dollar.     What  will  he  receive  to  whom  he  owed  100  dollars  ? 

Q.  88.  If  ten  dollars  worth  of  bread  is  sufficient  for  a  family  of  8 
persons  4  months,  what  will  it  cost  them  a  year  for  bread  at  that  rate  ? 

Q.  89.  If  a  manufacturer  pays  daily  to  the  men  in  his  employ,  one 
dollar  and  one  quarter  apiece,  to  the  women  seventy-five  cents  apiece, 
and  to  them  all  60  dollars  a  day,  how  many  men  and  women  does  ho 
employ,  provided  there  are  as  many  of  the  one  as  of  the  other  ? 

Q-  90.  Two  men  bought  a  barrel  of  flour  for  ten  dollars,  the  one 
paying  6  dollars,  and  the  other  4  dollars.  They  sold  the  flour  at  an 
advance  of  two  dollars.     What  part  of  the  gain  ought  each  to  have  ? 

Q.  91.  Three  men  bought  a  drove  of  cattle,  for  which  A  paid  400 
dollars,  B  600  dollars,  and  D  200  dollars.  They  lost  300  dollars ; 
now  what  sum  ought  each  to  pay  to  make  good  the  total  loss  ? 

Q.  92.  Two  men  hired  a  pasture  for  48  dollars  ;  A  put  in  4  horses 
3  months,  and  B  18  calves  6  months,  with  the  understanding  that  3 


QUESTIONS    FOR    EXAMINATION.  303 

calves  should  be  considered  the  same  as  1  horse.    What  sum  ought 
each  to  pay  for  the  use  of  the  pasture  1 

Q.  93.  A  man  and  his  wife  consumed  in  2  months  a  barrel  of  flour, 
but  when  the  husband  was  gone,  a  barrel  of  flour  lasted  the  wife  5 
months  ;  what  part  did  both  consume  in  one  month  1  What  is  the 
difference  between  what  they  both  consumed  and  what  the  wife  con- 
sumed'? How  many  pounds,  then,  did  each  consume  in  the  two 
months  T 

Q.  94.  A  man  having  a  horse  and  cow,  found  that  3  loads  of  hay 
would  keep  them  both  6  months,  and  when  he  had  no  cow,  the  same 
quantity  would  last  his  horse  10  months.  How  long  would  it  take 
each  to  consume  1  ton  of  hay  1 

Q.  95.  Suppose  A  and  B  can  mow  a  certain  field  in  6  days,  and  with 
the  assistance  of  C  will  mow  the  same  field  in  4  days :  how  much  of  it 
could  A  and  B  mow  in  one  day  1  How  much  could  the  three  do  in 
one  day  1     How  long  would  C  be  in  doing  it  alone  1 

Q.  90.  If  A  and  B  together  can  build  a  boat  in  4  days,  and  with  the 
assistance  of  C  can  do  it  in  3  days,  in  what  time  would  each  do  it  alone  T 

Q.  97.  If  a  third  of  six  be  3,  what  will  the  fourth  of  20  be  1 

Q.  98.  If  12  apples  be  worth  as  much  as  20  pears,  and  3  pears  cost 
1  cent,  what  is  the  price  of  100  apples'? 

Q.  99.  If  a  staff  8  feet  long  casts  a  shade  10  feet  long,  what  is  the 
height  of  that  pole  which  casts  at  the  same  time  a  shade  25  feet  long"? 

Q.  100.  If  a  family  of  8  persons,  in  24  months,  spend  240  dollars, 
how  many  dollars  would  three  times  as  many  persons  spend  in  one 
fifth  part  of  the  time  1 

Q.  101.  If  5  men  build  a  wall  20  rods  long  in  8  days,  how  many  men 
will  it  take  to  build  a  wall  30  rods  long  in  4  days '? 

Q.  102.  If  A  can  do  a  piece  of  work  in  1  hour,  B  in  2  hours,  C  in  3 
hours,  and  D  in  4  hours,  in  what  time  will  they  all  do  it,  if  they  work 
together  1 

Q.  103.  What  is  Practice'?  226.  What  are  Duodecimals T  227. 
How  is  unity  divided  in  Duodecimals  ]  What  is  the  rule  for  Cross 
Multiplication?  229. 

Q.  104.  What  is  Involution?  230.  What  is  the  square  of  Si- 
cube  of  3  ? — biquadrate  of  2  ^ — 5th  power  of  1 0 1 — 6th  power  of  1 0  ? — 
second  power  off  1 — third  power  of  |1 — fourth  power  of  3j'? — fifth 
power  of  .2 1— fourth  power  of .  02  ]— third  power  of  .002  ]— of  .0002  ? 

Q.  105.  What  is  Evolution]  233.  What  is  a  roof?  What  is 
meant  by  the  extraction  of  roots  ]  What  are  rational  and  surd  mem- 
bers '?     What  are  the-  signs  of  roots '? 

Q.  106.  What  is  the  square  rooti  235.  Rule?  237.  Rule  for 
finding  the  mean  proportional  between  two  numbers  *?  240. 

Q.  107.  What  is  the  square  root  of  36?— of  81?— of  .25?— of 
.0025  ?— of.0144?— of  .000081 1 

Q.  108.  What  is  a  cube?  241:  Cube  root?  What  is  the  form  of 
a  cube?  Why  is  the  cube  root  of  any  number  equal  to  the  length  of 
either  side  of  a  cube  ?    What  is  the  rule  ?  243. 


304  ARltHMfitiC. 

Q.  109.  What  is  the  Cube  Root  of  216 1— of  l,728t— of  ^\1— of 
15^?— of  271— of  .027]— of  .008  1— of  .000027  ?— of  27,000? 
What  is  the  general  rule  for  extracting  the  roots  of  all  powers'!  246. 

Q.  110.  What  is  Alligation?  248.  Alligation  Medial?  Alligation 
Alternate?  Rule?  249.  Arithmetical  Progression?  25 L  Terms? 
Rules  ]     Geometrical  Progression  ?  254. 

Q.  111.  What  is  an  Annuity?  256.  How  is  the  amount  found  at 
simple  interest?  257.  How  at  compound  interest!  258.  How  is 
the  present  worth  found?  259.  How  the  annuity  which  any  sum 
Will  purchase  ? 

Q.  112.  What  is  Permutation  ?  260.  What  is  the  rule  for  finding 
the  different  changes  which  may  be  made  with  any  given  number  of 
things?  261. 

Q.  113.  What  is  Position?  261.  Single  Position ?  Double  Po- 
sition! 262.     Rule?  263. 

Q.  114.  What  is  Mensuration?  263.  What  is  an  angle?  De- 
scribe the  different  angles.  264.  How  is  each  formed  ?  Describe 
the  different  triangles.  To  what  is  the  square  of  the  longest  side  of 
a  right-angled  triangle  equal  ? 

Q.  115.  Describe  the  different  quadrilateral  figures.  266.  How 
are  they  formed  1  What  is  a  circle  ?  How  are  its  area,  circumfe- 
rence, diametier  and  radius  found  ?  268. 

Q.  116.  What  is  a  solid?  271.  Describe  the  different  solids. 
Rule  for  finding  the  solidity  of  each  ?  What  is  Gauging  ?  277. 
Rule  for  finding  what  weight  of  water  may  be  put  into  a  given  vessel  ? 
Rule  for  gauging  a  cask  ?  What  are  the  two  rules  for  finding  the 
tonnage  of  vessels?  278. 

Q.  117.  What  is  Exchange?  279.  Bills?  Set  of  Exchange  ? 
What  has  occasioned  a  difference  in  the  value  of  our  gold  coins  ? 
283.  What  is  the  value  of  our  eagle  ?  What  are  the  two  methods 
for  reducing  a  sum  in  sterling  money  to  federal  money  ?  285. 
Q.  118.  How  many  pounds  sterling  must  a  merchant  in  London  re 
mit  to  New- York  to  pay  a  debt  of  $800  ? 


CONTENTS. 


PART  FIRST 

Mental  Coarse, ...  3 

PART    SECOND. 

Simple  Numbers, 26 

Compound  Numbers, ...  73 

Vulgar  Fractions, ....  109 

Decimal  Fractions, l'*^ 

Currencies, .        ,  161 

Percentage, 163 

Stocks 1^6 

Commission,  

Insurance, 

Loss  and  Gain, ^^® 

Interest,         . ^^^ 

189 
Compound  Interest, 

Discount,       ........••••••       ^^ 

Banking, ^^^ 

„       .  197 

Equation, ,       .       .        • 

199 

Proportion,  

222 

Fellowship,  

PART    THIRD. 

226 

Practice, 

.        .  227 

Duodecimals, 


306  CONTENTS. 

Involution,      .  230 

Evolution, 233 

Alligation, 248 

Progression, 251 

Annuities,      ....  256 

Permutation,  ........  ....  260 

Position,         ....  .        ,  .....  261 

Mensuration, 263 

Gauging, 277 

Tonnage, 278 

Exchange, 279 

Miscellaneous  Examples, '    .       .       .  288 

Questions  for  Examination, 297 


fUHITBBSITT, 


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V  GRAMMAR.     English  Grammar   ou4he   Pko- 

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fur  "Primaiy  Schools  and  Academies.     By  }i 

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'arger  than  the  Practical  and  Mental,  designed  for  bcl^plars  advun- 

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he  above  work  ;  designed  for  teachers  Oi.:  ■ 

GAJ.LAUDET'S  MOT[:En'.«  PRIM  r.R.     To  teach  her  child  its 
1  ,nv.    ^n, )  v,Q^,  ify  pg3{| .  designed  also  for  th'j  lowcs'     •^---  -  •>  "  '  '  >ry 
1  new  plan.     B)*  Rev.  T.  H.  Gallaudet 


